Thank goodness Phoebe Waller-Bridge, the ridiculously talented 33-year-old creator, writer and star of the marvellous Fleabag, didn’t land the lead in Doctor Who.

For a brief period, the bookies made her the favourite to play the first female incarnation of the time lord (leaving aside Joanna Lumley’s turn in the 1999 Red Nose Day spoof, The Curse Of Fatal Death). There’s no doubt whatsoever that Waller-Bridge would have been wonderful in the role – as, I’m sure, will the woman who eventually landed it, Jodie Whittaker.

However, locking her into the Tardis for the next few years would probably have cast doubt over the second season of Fleabag, due next year.

It would also have compromised the future of Waller-Bridge’s latest project, the brilliant Killing Eve, which starts on RTE2 tonight with a double-bill of episodes.

Made by America, the commercial cable outfit jointly owned by the corporation and AMC, it definitely owes something to the likes of Fargo and Barry, which showed it’s possible to take suspense, extreme violence and dark, deadpan humour and whip them up into a satisfyingly cohesive whole.

However, Killing Eve, adapted from Luke Jennings’ Villanelle novels, takes it all a step further, playing by nobody’s rules but its own. It’s a flashy spy thriller. It’s a tense game of cat-and-mouse. It’s a pitch-black comedy. At times, it’s even a bit of an office sitcom.

It bends and blends genres in such a wild and risky way, sharply switching tone and changing gear at the drop of a one-liner or the thud of a dead body, that it’s impossible to classify.

The Eve of the title is Eve Polastri – marvellously played by Sandra Oh with faultless comic timing – an American expat in London who’s stuck in a drab, frustrating desk job as an MI5 analyst and married to decent but slightly dull teacher, Niko (a nice role for An Klondike’s Owen McDonnell).

Like Melissa McCarthy’s character in Spy, Eve, who longs for a more exciting job, is a bit frazzled and chaotic but smarter than most of the people she works for.

When various unsavoury types start turning up dead all over Europe at regular intervals – an Italian Mafia boss here, a corrupt Russian politician there – Eve believes the killings are the work of a single female assassin whose modus operandi is that she doesn’t have one.

Needless to say, her superiors don’t believe her, and needless to say, Eve is right.

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The culprit is Villanelle (Jodie Comer), an elite, Paris-based contract killer who uses a variety of gruesomely inventive methods: a hairpin in the eye, a lethal injection, a toxic homemade fragrance and sometimes a good old-fashioned gun.

As every killing is different, there’s no trail of clues to follow.

It’s clear from the outset that Villanelle is a psychopath who kills as much for satisfaction as for money, hanging around after a hit so she can relish watching the light dim in her victim’s eyes.

She should be a repellent character, but thanks to Comer’s colourful, charismatic and sometimes very funny performance and Waller-Bridge’s fantastically inventive script she’s the most compelling twisted killer since Hannibal Lecter. You can’t wait to see what she does next.

Eve is sacked by MI5 for insubordination. Luckily, her potential has been spotted by MI6 hotshot Carolyn Martens (Fiona Shaw, terrific), who sets her up in an off-the-books operation with her own little team, including her former – and also now unemployed – boss, played by the always excellent David Haig, to follow up the case.

A chance encounter in a hospital toilet means Eve knows what Villanelle looks like, and Villanelle knows Eve is looking for her. 

That’s when the fun and games (and thrills) really begin.

To borrow a line Westworld borrowed from Romeo and Juliet, it’s a violent delight.

Killing Eve, RTE2, 9.30PM