  Contact homology for Legendrian submanifolds in standard contact
$(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks
with Lagrangian boundary conditions in complex $n$-space. It provides new
invariants of Legendrian isotopy. Using these invariants the theory of
Legendrian isotopy is shown to be very rich. For example, infinite families of
pairwise non-isotopic Legendrian $n$-spheres and $n$-tori, which are
indistinguishable by means of previously known invariants, are constructed.
  In a sense, the definition of contact homology presented in this paper is a
high dimensional analog of the work of Chekanov and others on Legendrian
1-knots in 3-space.
