  This paper deals with various cases of resonance, which is a fundamental
concept of science and engineering. Specifically, we study the connections
between periodic and unbounded solutions for several classes of equations and
systems. In particular, we extend the classical Massera's theorem, dealing with
periodic systems of the type \[ x'=A(t)x+f(t) \,, \] and clarify that this
theorem deals with a case of resonance. Then we provide instability results for
the corresponding semilinear systems, with the linear part at resonance. We
also use the solution curves developed in [8], [9] to establish the instability
results for pendulum-like equations, and for first-order periodic equations.
