  We study a class of systems of functional equations closely related to
various kinds of integrable statistical and quantum mechanical models. We call
them the finite and infinite Q-systems according to the number of functions and
equations. The finite Q-systems appear as the thermal equilibrium conditions
(the Sutherland-Wu equation) for certain statistical mechanical systems. Some
infinite Q-systems appear as the relations of the normalized characters of the
KR modules of the Yangians and the quantum affine algebras. We give two types
of power series formulae for the unique solution (resp. the unique canonical
solution) for a finite (resp. infinite) Q-system. As an application, we
reformulate the Kirillov-Reshetikhin conjecture on the multiplicities formula
of the KR modules in terms of the canonical solutions of Q-systems.
