We study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value. We show that they enjoy almost optimal efficiency estimates. We present several variants, establish their efficiency estimates and discuss possible implementations. In particular, their projection subproblems may be solved inexactly via relaxation methods, thus opening the way for parallel implementations. We discuss accelerations of relaxation methods based on simultaneous projections, surrogate constraints and conjugate and projected (conditional) subgradient techniques.