We report our recent results in the development of theoretically supported scalable algorithms for the solution of large scale complex contact problems of elasticity The algorithms combine the TFETI based domain decomposition method adapted to the solution of 2D and 3D frictionless multibody contact problems of elasticity with our in a sense optimal algorithms for the solution of the resulting quadratic programming problems Rather surprisingly, the theoretical results are qualitatively the same as the classical results on scalability of FETI for the linear elliptic problems The efficiency of the method is demonstrated by the results of numerical experiments with parallel solution of both coercive and semicoercive 2D and 3D contact problems.