Barbican Centre, London

Tube:
Barbican: 4 min walk; Hammersmith & City, Metropolitan, Circle lines
Moorgate: 7 min walk; Hammersmith & City, Metropolitan, Circle lines
St Paul's: 11 min walk; Central line

The nearest tube station served by the Night Tube (Friday and Saturday) is St Paul's on the Central line.
Bus:

Bus Route 153 runs directly past the Barbican along Chiswell Street. Alight at 'Silk Street'. 
The Barbican is also close to Bus Route 4 and 56 which serve Barbican tube station. 
Train:
The nearest train stations are Liverpool Street and Farringdon.

Liverpool Street: 11 min walk
Farringdon: 11 min walk

Old Street and Moorgate are also served by Great Northern and Thameslink trains and are within walking distance of the Barbican.