We describe a class of non-convolution type integral operators, where the integration is with respect to parameteis of special functions. This class includes the famous Kontorovich-Lebedev. Mehler-Fock, Olevskii, Fourier-Jacobi transformations, which are quite important, for instance, in the solutions of boundary value problems in tin; mathematical theory of elasticity. Our techniques involve Mellin-Barnes type representations of the kernels, and the Mellin transform. General boundedness conditions and Parseval equalities are established. A series of examples are presented.