This summer arts venues across London are competing for attention as the Olympic spotlight looms over us.

But while the torch zooms around the country and millions of people join their local gym, spurred on by the image of Jessica Ennis’s washboard tummy emblazoned on the side of a bus, I have been discovering a new corner of the city at Theatre Local in Peckham. It was quite a novelty to watch some exciting new plays produced by the Royal Court without worrying that I might bump into someone from Made in Chelsea.

Theatre Local is a scheme that aims to take plays to alternative spaces and reach a wider, more diverse audience. The project is located in the Bussey Building, a cultural hub sitting a stone’s throw away from Peckham Rye station, an area that is increasingly frequented by trendy young folk. Theatre Local is not only a performance space but also an arts venue with a cafe, offering free playwriting workshops and discussions, music, art and dance workshops. It even has a roof terrace where I watched the Shard’s admittedly underwhelming laser light show.

The first production in this year’s Theatre Local season was Belong by Bola Agbaje, which questions the notion of home, spotlighting the experience of an MP who flees to Nigeria to escape the political heat in Britain only to get caught in further tensions. The audience was enthralled and amused, judging by the laughter and nods of agreement, by Agbaje’s depiction of Nigerian culture, particularly the bustling market-scene where the cast of seven actors effortlessly created the mayhem of a busy street. Since Bola Agbaje is herself a British-Nigerian and former Peckham resident, the Bussey Building seemed like the ideal location for this satirical new play.

Vera Vera Vera by Hayley Squires was originally developed as part of the 2012 Young Writers Festival and transferred to Peckham after a run at the Royal Court earlier this year. A darkly comic play, Vera is hugely touching and compassionate, capturing the turmoil of a family devastated by the grief of losing a loved one; a remarkable achievement given that it is Squires’ writing debut. A tale of uplifting young love is juxtaposed with a dysfunctional family to create a poignant and believable play that gripped me from start to finish. Even a group of loudly sceptical schoolchildren were wowed into an appreciative silence, except for one overexcited boy who burst out with “She’s pissed off!” as one actor turned away from another. An accurate observation that, rather than prompting me to turn grumpily around in my seat, simply made me appreciate this child’s emotional investment in the characters.

Perhaps the most appealing aspect of Theatre Local is its affordable cost, aimed at attracting a local crowd to the see the plays. Audience members can book a ticket in advance for £10 or pay what they like on the door, as Hayley Squires has warmly stated on Twitter: “we ain’t precious about money, we just want ya to see it”.