We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $leq 11$. For dimensions $D=10,11
$ we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry algebra.
