  In this work we prove the lower bound for the number of $T$-periodic
solutions of an asymptotically linear planar Hamiltonian system. Precisely, we
show that such a system, $T$-periodic in time, with $T$-Maslov indices
$i_0,i_\infty$ at the origin and at infinity, has at least $|i_\infty-i_0|$
periodic solutions, and an additional one if $i_0$ is even. Our argument
combines the Poincar\'e--Birkhoff Theorem with an application of topological
degree. We illustrate the sharpness of our result, and extend it to the case of
second orders ODEs with linear-like behaviour at zero and infinity.
