Hochschild Homology of Twisted Tensor Products^*

We compute the Hochschild homology of some twisted tensor products of algebras, which are a natural generalization of the Ore extensions. We apply our result to the ring D_Q,P(X},∂/∂X) of differential operators of the multiparametric affine space, the ring 𝒪_q(spℂ^2v) of coordinates of the quantum symplectic 2v-dimensional space and the ring 𝒪_q(soℂ^2v) of coordinates of the quantum 2v-dimensional Euclidean space.