  We investigate the geometry of Legendrian complex projective manifolds
$X\subset\PP V$. By definition, this means $V$ is a complex vector space of
dimension $2n+2$, endowed with a symplectic form, and the affine tangent space
to $X$ at each point is a maximal isotropic subspace. We establish basic facts
about their geometry and exhibit examples of inhomogeneous smooth Legendrian
varieties, the first examples of such in dimension greater than one.
