Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.

Apart from the fact that its building is worth a visit by itself as it's the only purpose-built stable that functioned as a horse hospital within Central London that is still intact, it also offers some of the best underground and avant-garde culture in London. Aside from exhibitions, film screenings and music events it also houses 'The Contemporary Wardrobe Collection' which features vintage street fashion, couture items and accessories to the Film, TV and Fashion industries.

6a architects transformed the six-storey building at the corner of Redchurch Street into a multidisciplinary space. A shop, a gallery, a restaurant. 

In 1936 Oswald Mosley's British Union of Fascists attempted to march through East London in what was an attempt at an intimidating show of strength. Like today, this area was home to a large number of ethnic and religious minorities, particularly Jews. 300,000 people came out to oppose the march and blocked the route. The battle that followed was actually between the protesters and the police who tried to clear the route so that the march could take place. Seeing that they faced a losing battle and possibly a riot Mosley called off the march. The artist Dave Binnington began this mural in 1976 to commemorate that day, and it was eventually finished in 1982. The mural and the battle of Cable Street are both perfect examples of Britain at its very best.

Small restaurant on Highbury Corner that started off focusing on (basically) posh kebabs, but which is now producing some of London's most innovative dishes, focused around unconventional meats, cuts, and breads. Not often you can eat incredible food while listening to heavy rock and metal, but Lee Tiernan has nailed a winning formula.