The next generation of Cuban baseball stars may have a much easier time reaching the major leagues.

Cubans can sign under rules similar to what players from Japan, South Korea and Taiwan face, according to an agreement among Major League Baseball, its players’ association and the Cuban Baseball Federation announced Wednesday. Players from Cuba would be allowed to sign big league contracts without defecting.

For decades, Cuban players have arrived under mysterious circumstances, and stories have surfaced of the harrowing lengths they’ve gone to come play in the United States. Here are a few prominent examples:

Garvey arrives

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Before Cuban stars like Yasiel Puig and Yoenis Cespedes were even born, Barbaro Garbey made his major league debut for the World Series-winning Detroit Tigers in 1984. He’d come over during the 1980 Mariel boatlift, when some 125,000 Cubans were allowed to flee the island.

A quarter-century later, Garbey told USA Today he had to try multiple times to join the boatlift.

“The first three times, they recognized me, and said it was not for me,” he said. “The fourth time, the guy still recognized me, and he said, ‘OK, you want to go, get the hell out of here.’”

National teamers

The influx of Cuban defectors really began in 1991 when pitcher Rene Arocha, in the United States with the Cuban national team, walked away from teammates at a Miami airport. Slick-fielding shortstop Rey Ordonez scaled a fence when he left the Cuban team at the 1993 World University Games in Buffalo, New York.

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Pitcher Rolando Arrojo was in Georgia with the Cuban Olympic team in 1996 when he sneaked out of a hotel, jumped in a vehicle and was off to Miami. He became an All-Star for Tampa Bay in 1998.

The Hernandez’ brothers

The Cuban national team was training in Mexico in 1995 when Livan Hernandez walked off from a hotel and was able to take a flight to Venezuela. Two years later, he was the MVP of the World Series for the Florida Marlins.

Orlando Hernandez, Livan’s half-brother, was banned from Cuban baseball after Livan’s departure. Orlando was able to escape with a small group via boat in December 1997. He made it to the Bahamas and eventually joined the Yankees. He was part of three World Series winners in New York, and also won a World Series with the White Sox in 2005.

Puig’s journey

Puig defected from Cuba in 2012 after several failed attempts. He was smuggled out of Cuba by traffickers linked to a Mexican drug gang, according to court testimony.

The Los Angeles Dodgers’ star said in a statement Wednesday that he is happy to know future Cuban players won’t have to go through that type of ordeal.

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Defection Ordeals May Be a Thing of the Past for Cuban Stars