These lectures are devoted to study of separability, entropy and channels in infinite dimensional Hilbert spaces. The first two sections deal with criteria of compactness for subsets in state space and collections of state ensembles. Next a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces is given and an example of separable state that is not countably decomposable is provided. The structure theorem for the quantum communication channels that are entanglement-breaking is proved, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. We then discuss (dis)continuity properties of the quantum entropy and relative entropy in the infinite dimensional case. Sufficient conditions for the continuity of the output entropy of a quantum channel and related important entropic quantities are given. In case of compact constraints conditions for existence of optimal ensembles achieving the χ-capacity are given. These results are then applied to Gaussian channels.