  A new type of classical billiard - the Andreev billiard - is investigated
using the tangent map technique. Andreev billiards consist of a normal region
surrounded by a superconducting region. In contrast with previously studied
billiards, Andreev billiards are integrable in zero magnetic field, {\it
regardless of their shape}. A magnetic field renders chaotic motion in a
generically shaped billiard, which is demonstrated for the Bunimovich stadium
by examination of both Poincar\'e sections and Lyapunov exponents. The issue of
the feasibility of certain experimental realizations is addressed.
