There is one name which is defining this spring and its Derek Jarman. See his films all across London including The Gate Cinema in Notting Hill and Picturehouse cinemas; go to the exhibition at the Serpentine curated by film artist Isaac Julien (until 13 April); see the films in the Lightbox space at Tate Britain (5 April to 1 June). http://www.serpentinegallery.org/2007/04/derek_jarman_curated_by_isaac.html
Coming of Age: American Art 1850s to 1950s is on now at the Dulwich Picture Gallery until 8 June.
http://www.dulwichpicturegallery.org.uk
The Courtauld Institute's 8th East Wing Collection is on now, displayed throughout the academic areas, featuring Mark Wallinger, Antony Gormley, and lots of no doubt more interesting new artists.
www.eastwing8.co.uk
Contemporary painter Natasha Kissell's show The Hours is on at Eleven Fine Art from 26 March to 26 April. http://www.elevenfineart.com
Go to Tate Britain and see the Neoclassical Sculpture; or rather, don't look that close but just absorb the atmosphere of the darkened Duveen galleries; and see the greta Peter Doig exhibition.
www.tate.org.uk/britain
There is an exciting season of films on at Tate Modern as part of two concurrent themes: Paradise Now! Essential French Avant-garde Cinema 1890-2008 (from Dada and Duchamp to May 68 and contemporary artists), and 1, 2, 3...Avant-garde Film, Art Between Experiment and Archive (Polish avant-gardes over the last 80 years). Other film events include Warhol's Blow up (10 April) and Cindy Sherman's Office Killer (11 May).
http://www.tate.org.uk/modern/eventseducation/film/
Also on now is the latest Leve 2 Gallery show: Here We Dance.
Coming up in May at Tate Modern is an interesting-sounding photography show called Street & Studio: An Urban History of Photography, which features Diane Arbus, Beaton, Cartier-Bresson, Mapplethorpe, Tillmans and many more. 31 May Cornelia Parker will talk about her work at TM's Starr Audiorium

