  We review the solutions of O(N) and U(N) quantum field theories in the large
$N$ limit and as 1/N expansions, in the case of vector representations. Since
invariant composite fields have small fluctuations for large $N$, the method
relies on constructing effective field theories for composite fields after
integration over the original degrees of freedom. We first solve a general
scalar $U(\phib^2)$ field theory for $N$ large and discuss various
non-perturbative physical issues such as critical behaviour. We show how large
$N$ results can also be obtained from variational calculations.We illustrate
these ideas by showing that the large $N$ expansion allows to relate the
$(\phib^2)^2$ theory and the non-linear $\sigma$-model, models which are
renormalizable in different dimensions. Similarly, a relation between $CP(N-1)$
and abelian Higgs models is exhibited. Large $N$ techniques also allow solving
self-interacting fermion models. A relation between the Gross--Neveu, a theory
with a four-fermi self-interaction, and a Yukawa-type theory renormalizable in
four dimensions then follows. We discuss dissipative dynamics, which is
relevant to the approach to equilibrium, and which in some formulation exhibits
quantum mechanics supersymmetry. This also serves as an introduction to the
study of the 3D supersymmetric quantum field theory. Large $N$ methods are
useful in problems that involve a crossover between different dimensions. We
thus briefly discuss finite size effects, finite temperature scalar and
supersymmetric field theories. We also use large $N$ methods to investigate the
weakly interacting Bose gas. The solution of the general scalar $U(\phib^2)$
field theory is then applied to other issues like tricritical behaviour and
double scaling limit.
