  Generalized Heisenberg algebras $\H(f)$ for any polynomial $f(h)\in\C[h]$
have been used to explain various physical systems and many physical phenomena
for the last 20 years. In this paper, we first obtain the center of $\H(f)$,
and the necessary and sufficient conditions on $f$ for two $\H(f)$ to be
isomorphic. Then we determine all finite dimensional simple modules over
$\H(f)$ for any polynomial $f(h)\in\C[h]$. If $f=wh+c$ for any $c\in \C$ and
$n$-th ($n>1$) primitive root $w$ of unity we actually obtain a complete
classification of all irreducible modules over $\mathcal{H}(f)$. For many
$f\in\C[h]$, we also prove that, for any $n\in\N$, $\mathcal{H}(f)$ has
infinitely many ideals $I_n$ such that $\mathcal{H}(f)/I_n\cong M_n(\C)$, the
matrix algebra.
