  We investigate the problem of the realization of a given graph as the Reeb
graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$
with finitely many critical points, where $M$ is a closed manifold. We show
that for any $n\geq2$ and any graph $\Gamma$ admitting the so called good
orientation there exist an $n$-manifold $M$ and a Morse function $f\colon
M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is
isomorphic to $\Gamma$, extending previous results of Sharko and
Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize
the number of cycles. Furthermore, we provide a complete characterization of
graphs which can arise as Reeb graphs of surfaces.
