  States of a quantum mechanical system are represented by rays in a complex
Hilbert space. The space of rays has, naturally, the structure of a K\"ahler
manifold. This leads to a geometrical formulation of the postulates of quantum
mechanics which, although equivalent to the standard algebraic formulation, has
a very different appearance. In particular, states are now represented by
points of a symplectic manifold (which happens to have, in addition, a
compatible Riemannian metric), observables are represented by certain
real-valued functions on this space and the Schr\"odinger evolution is captured
by the symplectic flow generated by a Hamiltonian function. There is thus a
remarkable similarity with the standard symplectic formulation of classical
mechanics. Features---such as uncertainties and state vector reductions---which
are specific to quantum mechanics can also be formulated geometrically but now
refer to the Riemannian metric---a structure which is absent in classical
mechanics. The geometrical formulation sheds considerable light on a number of
issues such as the second quantization procedure, the role of coherent states
in semi-classical considerations and the WKB approximation. More importantly,
it suggests generalizations of quantum mechanics. The simplest among these are
equivalent to the dynamical generalizations that have appeared in the
literature. The geometrical reformulation provides a unified framework to
discuss these and to correct a misconception. Finally, it also suggests
directions in which more radical generalizations may be found.
