  We present a systematic theoretical study of the BCS-BEC crossover in
two-dimensional Fermi gases with Rashba spin-orbit coupling (SOC). By solving
the exact two-body problem in the presence of an attractive short-range
interaction we show that the SOC enhances the formation of the bound state: the
binding energy $E_{\text B}$ and effective mass $m_{\text B}$ of the bound
state grows along with the increase of the SOC. For the many-body problem, even
at weak attraction, a dilute Fermi gas can evolve from a BCS superfluid state
to a Bose condensation of molecules when the SOC becomes comparable to the
Fermi momentum. The ground-state properties and the
Berezinskii-Kosterlitz-Thouless (BKT) transition temperature are studied, and
analytical results are obtained in various limits. For large SOC, the BKT
transition temperature recovers that for a Bose gas with an effective mass
$m_{\text B}$. We find that the condensate and superfluid densities have
distinct behaviors in the presence of SOC: the condensate density is generally
enhanced by the SOC due to the increase of the molecule binding, the superfluid
density is suppressed because of the non-trivial molecule effective mass
$m_{\text B}$.
