Tender, remarkably ingenious and scalp-pricklingly scary, It Follows begins, in classic Halloween fashion, with a teen girl in the suburbs fleeing in utter panic from her front door. She's being chased, very slowly, by something we can't see. In fact, no one can see it, except for her and others like her – the victims of a sexually transmitted hex that’s sent a shape-sifting succubus implacably into their lives.

The premise, written thus, sounds pretty hokey. It's not at all. The movie's a brilliantly fresh spin on a classic model – the pass-on-the-curse conceit which horror fans will know from MR James's shivery short story Casting the Runes, and its numerous cinematic offspring, from Jacques Tourneur's Night of the Demon (1957) to the Japanese cult hit Ringu (2000) and its American remake, The Ring (2002).

It Follows – which deserves even more marks for that marvellously suggestive title – does this entire lineage proud, not just by switching tacks from runic subterfuge or videotape circulation to the rather Cronenbergy gambit of inflicting a demon on your unfortunate sex partner. It's altogether smart, subtextually fascinating, and more or less a contemporary horror fan's dream come true.

The genre trappings make it a nifty, commercially tantalising venture for its writer-director, David Robert Mitchell, after his terrific first feature in 2010, a too-little-seen reverie of awkward teen yearning called The Myth of the American Sleepover. He matches that film’s subtle sadness and complete sincerity here, while furthering its already substantial ambition, essentially by dropping a terrifyingly unhurried and many-faced stalker right into the middle of it.

Giving this revenant, or whatever the hell it is, the Michael-Myers-in-Halloween constriction of only walking at all times constitutes both a neat Carpenter homage and a clammy-palmed atmospheric masterstroke. It’s just one of the rules Mitchell sets himself and sticks to. He never cheats, never stretches the premise for cheap shocks. They’re utterly earned, and had the Cannes critics’ week audience jumping – in my case swearing, on at least one occasion - with helpless fright.

Going in, you don’t want to know too much about this, in plot and character terms: it’s possible the trap set for these experimenting youngsters might grip less if you were already in on it. Still, it’s worth counting all the many ways sex functions in the film – as weapon, succour, self-defence, straightforward release – and how they map on metaphorically to the ways it’s always used.

Mitchell wields his deadly external threat, not just as an even bigger incentive to cop off than standard teenage hormones, but as the strange opposite, too – as an invitation to pause, to stop in your tracks, and think about morality and consequence. It can’t always be a terrific experience, being used for sex as a compliant virgin, say. But Mitchell supplies the creepiest ulterior motives since Scarlett Johansson hit the Glasgow club scene in Under the Skin. We get a predator that’s visible and not, recognisable and foreign-bodied, as single-minded as the Terminator and as fetidly human as that rotting old lady in The Shining. This, the ‘It’ of It Follows, springs from the same It teenagers are generally craving and constantly thinking about. It’s something nasty in the woodshed. It’s a blind date to make your blood run cold.