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Well, the capital is the obvious place to start, and aside from the massive converted power station that is Tate Modern, there are interesting permanent collections at Tate Britain in Pimlico, the National Gallery on Trafalgar Square - and if heading to the latter you may as well pop into the National Portrait Gallery next door too (which is hosting a Lucien Freud retrospective until May 27).
The Saatchi Gallery in Chelsea specialises in modern art and all exhibitions are free to enter - and other noteworthy venues include The Royal Academy of Arts, the Hayward Gallery, the ICA and the Serpentine Gallery.
There are many, many more galleries open to the public also - and magazine Time Out has a good listings section.
Both Glasgow and Edinburgh have strong galleries dedicated to modern art, with the Scots' capital boasting the Scottish National Gallery of Modern Art and its auld rival having its own impressive Gallery of Modern Art (GoMA) - along with independent venue The Modern Institute.
Dundee has its own modern art venue (Dundee Contemporary Arts).
As the name suggests, the Baltic Centre for Contemporary Art in Gateshead specialises in modern art and played host to the Turner Prize last year.
Leeds Art Gallery and the adjoining Henry Moore Institute have a strong collection of 20th century art - bolstered by the support of the famous sculptor.
Manchester Art Gallery works some interesting contemporary pieces into its permanent collection and The Lowry in nearby Salford specialises in contemporary art alongside the work of the Lancastrian painter.
Tate Liverpool at the city's Albert Dock has an interesting schedule which includes a sculpture exhibition until April 1 and a room installation by Martin "lightbulb" Creed until May 27.
Sheffield has a strong art scene and a dedicated modern art venue in the shape of the Site Gallery. The Yorkshire Sculpture Park, near Wakefield, is also well worth a visit on a nice day - featuring works by the likes of Sir Henry Moore and Andy Goldsworthy.
The Midlands and South East
Birmingham's Ikon Gallery is dedicated to contemporary art, with exhibits currently arranged over two floors in a former school. A new purpose-built gallery is in the pipleine for 2020 though.
The New Art Gallery in Walsall has a strong collection of 20th century art and represents current artists well, while the Nottingham Contemporary also holds exhibitions of modern art in a striking building.
Modern Art Oxford hosts regular exhibitions from modern artists while the same city's Ashmolean Museum has a nice little collection of modern works.
Down on the coast in (sometimes) sunny Kent the Turner Contemporary has brought modern art to the seaside town of Margate - which has an exhibition of JMW Turner's sensational paintings until May 13, but usually shows more modern work.
The National Museum Cardiff is strongest on impressionism but is worth checking out for modern work too. The city also houses the Chapter Arts Centre, which has regular contemporary exhibitions.
The Attic Gallery and Glynn Vivian Art Gallery in Swansea are also good for 20th century and 21st century work.
The MOMA (Museum of Modern Art) Wales in the alternative lifestyle enclave of Machynlleth has a constantly changing line-up of work from modern Welsh artists.
The South West
Painters have long flocked to St Ives in Cornwall like seagulls to a discarded bag of chips, and since 1982 art buffs have enjoyed feeding off the fine displays of contemporary art at the Tate St Ives.
The council-run Bristol Museum and Art Gallery has an interesting modern art section, and the trendy city also hosts to a number of smaller galleries and arts centres.
Have we missed anywhere important? Let us know in the comments below...