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One in, one out: it's a rule that works for busy nightclubs, not generally for life. But this heartwarming two-hander tackles both ends of life's journey, as central character Leanne says goodbye to her ageing grandmother and sees the birth of her own longed-for baby, too. It's a gently humorous and very watchable family story, with a few stand-out emotional scenes.

Leanne Mckenzie – both the actor and the character are called Leanne – brings a deft balance to the role: outwardly confident and even brash, but insecure and fretful when it comes to the baby inside her. Katie Lambert plays Leanne's partner, narrating the story with crisp practical efficiency and providing a counterpoint to Leanne's occasional panic. (Yes, they're a lesbian couple, and yes, a turkey-baster is involved. But nobody in the play makes much of a fuss about that, and nor will I.)

The star of the show, though, is Leanne's nan Ella, also played by Mckenzie. Ella is close to death, and knows it; cleverly though, it's not her passing she wants to talk about, but her memories of people she has lost. Mckenzie's characterisation of Mrs Mackay as a Scottish matriach is a genuine delight: formidable, straight-talking, yet loving. Her description of her daughter's funeral is particularly moving, delivered with sensitivity and commitment by Mckenzie.

Ella's life has been long, and the script offers a genuine feeling for the scale of the changes she's witnessed: from poverty and slum clearances, to the pride of bringing up a family. There's a strong sense of place as well – the place being Edinburgh, a city I happen to know well. In fact, the richness of detail and realism of the anecdotes leave me wondering if this piece is at least partly autobiographical.

In one sense that's a big compliment, but in another it hints at a problem. Wan In Wan Oot doesn't have the clearest of through-lines; there are some significant plot points, but there are also quite a lot of detours. Real life's like that – but theatre generally isn't. There are missed opportunities here to amplify the impact of each moment, by linking it into a clearer theme.

Near the end, for example, Katie's very English upbringing provides an unexpected way to break the ice with her Scottish Catholic in-laws. That's charming and affecting, but it could be genuinely moving if the ground were prepared earlier on. Perhaps they could emphasise the cultural gulf between the families, or at least have Katie reflect on this particular side of her childhood.

As it is, we have a lot of touching and very believable anecdotes, but they never quite coalesce into something that's more than the sum of its parts. Still, Wan In Wan Oot is a finely-observed character piece, and a rewarding window into a richly-textured slice of family life.