Recent experiments on conduction between a semiconductor and a superconductor have revealed a variety of new mesoscopic phenomena. Here is a review of the present status of this rapidly developing field. A scattering theory is described which leads to a conductance formula analogous to Landauer's formula in normal-state conduction. The theory is used to identify features in the conductance which can serve as "signatures" of phase-coherent Andreev reflection, i.e. for which the phase coherence of the electrons and the Andreev-reflected holes is essential. The applications of the theory include a quantum point contact, quantum dot, weak localization, universal conductance fluctuations, shot noise, and reflectionless tunneling. This review is based on lectures at the Les Houches summer school, Session LXI, 1994, to be published in: Mesoscopic Quantum Physics, E. Akkermans, G. Montambaux, and J.-L. Pichard, eds. (North-Holland, Amsterdam).