  We consider the Brascamp--Lieb inequalities concerning multilinear integrals
of products of functions in several dimensions. We give a complete treatment of
the issues of finiteness of the constant, and of the existence and uniqueness
of centred gaussian extremals. For arbitrary extremals we completely address
the issue of existence, and partly address the issue of uniqueness. We also
analyse the inequalities from a structural perspective. We obtain two new
proofs of Lieb's fundamental theorem concerning exhaustion by gaussians. Our
techniques are partly based upon monotonicity formulas for positive solutions
to heat equations in linear and multilinear settings.
