  We present a Ginzburg-Landau theory for a two-band superconductor with
emphasis on MgB$_{2}$. Experiments are proposed which lead to identification of
the possible scenarios: whether both $\sigma $- and $\pi $-bands superconduct
or $\sigma $% -alone. According to the second scenario a microscopic theory of
superconducting MgB$_{2}$ is proposed based on the strongly interacting $%
\sigma $-electrons and non-correlated $\pi $-electrons of boron ions. The
kinematic and Coulomb interactions of $\sigma $% -electrons provide the
superconducting state with an anisotropic gap of $s$% *-wave symmetry. The
critical temperature $T_{c}$ has a non-monotonic dependence on the distance $r$
between the centers of gravity of $\sigma $- and $\pi $-bands. The position of
MgB$_{2}$ on a bell-shaped curve $T_{c}$ ($% r)$ is identified in the overdoped
region. The derived superconducting density of electronic states is in
agreement with available experimental and theoretical data. It is argued that
the effects of pressure are crucial to identify the microscopic origin of
superconductivity in MgB$_{2}$. Possibilities for $T_{c}$ increase are
discussed.
