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The topic of this book is of interest to contemporary mathematicians and has some interpretations in terms of modern mathematical physics. More precisely, the aspirant investigates certain cohomology of infinite dimensional Lie superalgebras which help to describe the deformations of certain structures related to these Lie superalgebras. Over the (1;N)-dimensional real superspace, we classify osp(Nj2)-invariant differential operators acting on the superspaces of weighted densities, where osp(Nj2) is the orthosymplectic Lie superalgebra. This result allows us to compute the first differential osp(Nj2)-relative cohomology of the Lie superalgebra K(N) of contact vector fields with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We classify generic formal osp(3j2)-trivial deformations of the K(3)-module structure on the superspaces of symbols of differential operators.