  This paper investigates the link between the Maximum Principle and the sign
of the (generalized) principal eigenvalue for elliptic operators in unbounded
domains. Our approach covers the cases of Dirichlet, Neumann, and (indefinite)
Robin boundary conditions and treat them in a unified way. For a certain class
of elliptic operators (including the class of selfadjoint operators), we
establish that the positivity of the principal eigenvalue is a necessary and
sufficient condition for the validity of the Maximum Principle. If the
principal eigenvalue is zero, no general answer holds; instead, under a natural
condition on the domain's size at infinity, we show that the operator satisfies
what we call the Critical Maximum Principle. We also address the question of
the simplicity of the principal eigenvalue, and a series of counterexamples is
proposed to disprove some possible misconceptions. Our main results are new
even for the more classical cases of Dirichlet boundary conditions and
selfadjoint operators.
