In the paper, we develop a composite version of Mirror Prox algorithm for solving convexconcavesaddle point problems and monotone variational inequalities of special structure, allowingto cover saddle point/variational analogies of what is usually called “composite minimization”(minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions“as if” there were no nonsmooth component at all). We demonstrate that the composite MirrorProx inherits the favourable (and unimprovable already in the large-scale bilinear saddle pointcase) O(1/ε) efficiency estimate of its prototype. We demonstrate that the proposed approach canbe successfully applied to Lasso-type problems with several penalizing terms (e.g. acting togetherl₁ and nuclear norm regularization) and to problems of semi-separable structures considered inthe alternating directions methods, implying in both cases methods with the O(1/ε) complexitybounds.