Jenny Gill was in New York’s Cornell Weill hospital earlier this year for the birth of her son, Jack, when a photographer stopped by to take snapshots of the mother and newborn. The practice is common in hospitals, but what the photographer did next surprised Gill.

“In the middle of taking the pictures, she pulls out this cutely wrapped onesie and says, ‘Oh, here’s a free Disney onesie. We’ll just need your email address,’” Gill recalls. “It weirded me out. I just gave birth, please lay off with the Disney already!”

Disney is unlikely to lay off anytime soon, and neither are the countless other brands in need of dollars. They’re part of a trend—fueled in part by the growth of digital devices—toward aggressively targeting a demographic that didn’t exist, in marketers’ eyes, until recently: infants to 3-year-olds. By getting their logos and iconic characters in front of babies—even those with still-blurry eyesight—they hope to establish brand-name preference before she or he has uttered a word.

Licensed characters, of course, have been plastered on packaged foods, toys, and every piece of clothing and accessory imaginable for decades—some reach back a century. But over the course of the past couple of years something has shifted. Now the strategy of marketing branded goods to tots has ramped up to the point that even high-end fashion brands are jumping in. Designers including Versace, Fendi, and Marc Jacobs have all recently released lines for kids barely walking, and Cynthia Rowley has even come out with designer diapers.

“It’s a segment where, particularly in the luxe category, it’s an attractive target,” says Marty Brochstein, senior vice president at the International Licensing Industry Merchandisers’ Association. “People like to dress up their kids and show off their kids. For that luxe brand itself…it’s beginning the relationship with the child.”

There are an awful lot of logos vying for that relationship. This summer, for instance, Maclaren, the stroller company, teamed up with Manhattan-based Dylan’s Candy Store to co-brand a brightly colored $150 stroller. Audi manufactures branded teddy bears.

“[Brands] are going younger and younger all the time,” says Dan Acuff, a former marketing consultant to Hasbro, Mattel, Nestlé, and others. “Babies don’t distinguish between reality and fantasy, so they think, ‘Let’s get them while they’re susceptible.’”

Martin Lindstrom, a brand consultant whose new book, Brandwashed, was released last week, says, “Marketers are getting more and more desperate to generate sales, so we’re slowly moving the ethical lines. When it comes to families, parents are much more influenced by the kids’ and babies’ preferences for brands.” And babies do have preferences. The Campaign for a Commercial-Free Childhood reports that by six months, babies are forming mental images of corporate logos and mascots, and that babies request brands as soon as they can speak. Other studies show that by the time an American child is 3 years old, he or she can recognize an average of 100 brand logos.

Texas A&M marketing professor James McNeal calls it the “drool factor.” Put your baby in a bib imprinted with an image of a branded character or in front of a video on a tablet, and voila: brand recognition in the crib.

The free Disney hospital goodie, in fact—a team effort with Our365, a company that sells bedside baby pictures—was no one-off, but an integral facet of Disney Baby, which launched early this year and targets infants in a more focused manner than past efforts. Disney has also been a leader in reaching the young set online and in the fast-growing sector of baby-geared apps. (Disney’s many such downloads include Toy Story Read-Along; Finding Nemo: My Puzzle Book; and Disney Princess Dress Up.) The Campaign for a Commercial-Free Childhood, which estimates that 14 percent of children under 2 years old spend more than two hours a day with screen media, believes marketers are subtly encouraging parents to soothe fussy babies with cell phone videos and games from sites that also sell a host of baby-targeted goods.

A recent study conducted by the Joan Ganz Cooney Center found that 80 percent of kids under the age of 5 use the Internet weekly, and 60 percent of kids 3 and younger are now watching videos online. It seems they’re taking their cue from their parents. For the past two years, Parenting magazine and BlogHer have asked mothers to weigh in on the ages they would permit their children to own specific technologies. In 2010, the average age they said children could have their own mobile phone was 13-years-old. In 2011, however, 1 in 4 mothers allowed their children to interact with a mobile phone by age 2.

“Our mouths dropped,” says BlogHer’s Jane Collins, director of market research. “I don’t think we expected to see such numbers of the 2- and 3-year-olds interacting with these devices. It seems like it progressed very rapidly. [And] the younger the mom, the more likely it is she’ll have children at a very young age experiencing technology.”

The official stance of the American Academy of Pediatrics is that babies shouldn’t have any time in front of a screen—none—before the age of 2. “There is no hurry in introducing your children to new technology,” says Vic Strasburger, chief of the academy’s division of adolescent medicine. “There’s no evidence that a baby given a computer is going to do better in school than a baby without one.” There are, however, piles of evidence that simply reading to your baby will accelerate his or her language acquisition. How analog.

But any parent who’s flown with a fussy infant can attest to the magical powers of digital devices. They’re handheld baby-sitters, digital pacifiers. It’s a parenting tactic as old as the television, only now these televisions are portable. Babies are especially enamored with smartphones, tablets, etc., given that a simple touch of a keyboard rewards them with shapes and sounds. (Intuitive devices like Apple’s iPad are especially alluring.) For proof, look no further than the hundreds of YouTube videos showing infants and toddlers smitten by such devices, to the obvious delight of their proud parents.

In addition to Disney, brands including Fisher-Price, Mattel, Sesame Street, Scholastic, National Geographic Kids, Nick Jr., LeapFrog, and TV characters like Teletubbies—almost every major baby and toddler brand with goods to sell—has an app to download and/or a website dedicated to online games, populated by branded characters and a click away from their online stores.

Infants now even have their own TV “network” with BabyFirstTV.com, which offers subscription-based content including videos, games, educational activities, and music. A 2-year-old can watch some content free, but to get more, parents have to pay—and if their toddler wants it, he or she will probably get it, thanks to what’s known in the business as the “nag factor.”

Marketers, meanwhile, are cooking up ever slicker and savvier gadgets. Brand new to the market is the Vinci, which debuted this month. It purports to be the first-ever educational handheld device for kids 0-4. The Vinci tablet looks like an oversized iPhone, runs on Android, and comes encased in a drool-proof bumper.

“We are converging three categories—technology, science, and education—to make a brand new learning system for the youngest children,” says Vinci’s inventor, Dan D. Yang, formerly a telecom entrepreneur. “Passively watching is definitely something we do not do. We’re talking about play-based learning under the control of the child with parental guidance.”

Out of the box (for $389 or $479, depending on the model), the Vinci is loaded with original interactive storybooks, virtual worlds for small fingers to explore, and no recognizable characters—as of yet. Parents will soon be able to plug the device into the Android app store and download branded apps to their hearts’ content.

Watchdog groups, and others, believe such marketing strategies can negatively affect a child’s well-being. “Bringing these kids into the market economy in such a wholesale way has raised all kinds of questions in terms of childhood development and socialization issues,” says Jeff Chester, founder and executive director of the Center for Digital Democracy. (He and his wife, Kathryn Montgomery, a professor in the School of Communication at American University, recently wrote a report, “Interactive Food and Beverage Marketing: Targeting Children and Youth in the Digital Age.”) “It’s not just about easy baby bucks; it’s about how kids play, how they socialize, cognitive development in terms of early brain behavior.”

In her Pulitzer Prize-winning novel A Visit From the Goon Squad, author Jennifer Egan paints a satirical picture of a dystopian future where pop music is marketed to pre-verbal consumers called “pointers” who make purchases with their own handheld devices. Is that where we’re headed?

“We’re raising a generation of children who are bored unless they’re in front of a screen,” adds Susan Linn, director of the Campaign for a Commercial-Free Childhood. “There’s no credible evidence that babies learn anything useful from screen media.” (In 2009 the organization sued Disney’s Baby Einstein over the claim that its “developmental and entertainment” videos and toys for babies and toddlers were educational. Linn and her group claimed victory when Disney offered refunds for Baby Einstein products, acknowledging that the line does not, in fact, make babies smarter as advertised.) Are marketers justified in targeting children 0-3 years old?

In a policy paper for the American Academy of Pediatrics, Strasburger, who’s a professor of pediatrics at the University of New Mexico School of Medicine, writes that “children under the age of 7 are psychologically defenseless against advertising.” In Sweden, in fact, all advertisements aimed at children under the age of 12 have been banned. In 1750 B.C. the Babylonian Code of Hammurabi made it a crime to sell anything to a child, punishable by death. (Granted, the ancient Babylonians also made it a crime for a son to strike his father, punishable by having his hands cut off.)

But lest you think it’s just hand-wringing watchdogs and helicopter parents doing the tongue clucking here, one of the hottest advertising executives of the last generation has also called for a moratorium. Alex Bogusky, founding partner of Crispin Porter + Bogusky and now founder of the consumer think tank FearLess Cottage, recalls passing a McDonald’s with his kids when they were toddlers. Without ever having been to the fast-food chain, his children (now 12 and 15) told him they loved it and wanted to go—and all because of the restaurant’s advertising, he says, which focused on the toys and Ronald.

“Probably most of us have been in the car and had your kids be, like, ‘I want to go to McDonald’s, I want to go to Burger King,’ wherever. And basically you get in a fight over it,” he tells Adweek. “I don’t think that consumers are thinking it through. The fight you’re getting in is between you, your kid, and some MBA who has developed a relationship and a conversation with your child. I don’t know why we feel like it would be such a mistake to say, ‘You know what? We won’t advertise to kids.’”

But targeting younger and younger kids, as can be seen by the dollars pouring into the coffers of these brands, does not bother all parents.

“Although I’m not a huge fan of buying my kids products drenched in licensed characters, if it makes my life easier as a parent, then I’m all for it,” says Kristen Chase, an Atlanta, Ga.-based mother of four and co-editor of the popular parenting blogs Cool Mom Picks and Cool Mom Tech. “I owe Transformers underpants to tear-free potty training. And Dora bandages seem to make boo-boos better much quicker than the plain ones.”

Julie Robichaux, a Montpelier, Vt., mother of two who runs the blog A Little Pregnant, refuses to get too worked up over the fight for her kids’ attention. “Until someone sends my 3-year-old a credit card and takes him shopping without me,” she says, “I don’t much care one way or the other.”