  We systematically study topological phases of insulators and superconductors
(SCs) in 3D. We find that there exist 3D topologically non-trivial insulators
or SCs in 5 out of 10 symmetry classes introduced by Altland and Zirnbauer
within the context of random matrix theory. One of these is the recently
introduced Z_2 topological insulator in the symplectic symmetry class. We show
there exist precisely 4 more topological insulators. For these systems, all of
which are time-reversal (TR) invariant in 3D, the space of insulating ground
states satisfying certain discrete symmetry properties is partitioned into
topological sectors that are separated by quantum phase transitions. 3 of the
above 5 topologically non-trivial phases can be realized as TR invariant SCs,
and in these the different topological sectors are characterized by an integer
winding number defined in momentum space. When such 3D topological insulators
are terminated by a 2D surface, they support a number (which may be an
arbitrary non-vanishing even number for singlet pairing) of Dirac fermion
(Majorana fermion when spin rotation symmetry is completely broken) surface
modes which remain gapless under arbitrary perturbations that preserve the
characteristic discrete symmetries. In particular, these surface modes
completely evade Anderson localization. These topological phases can be thought
of as 3D analogues of well known paired topological phases in 2D such as the
chiral p-wave SC. In the corresponding topologically non-trivial and
topologically trivial 3D phases, the wavefunctions exhibit markedly distinct
behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap
functions are included in the dynamics, the SC phases with non-vanishing
winding number possess non-trivial topological ground state degeneracies.
