The Design Museum is a museum in Kensington, next to Holland Park, which covers product, industrial, graphic, fashion and architectural design. There is also a nice cafe on the 1st floor with beautiful views of the park.
I love the architecture of the Barbican - it is so important to its historical context, at a time when London was being shaped for modernity after the war - they also aspired to make it a bold art centre. We need bold, always.
Tucked away down a cobbled alley behind Russell Square Station, The Horse Hospital, a converted stable, is a unique and totally independent arts space specialising in underground and experimental media. The gallery space itself (known as The Chamber of Pop Culture) houses unusual exhibitions, film screenings and other events.