  I describe the results for the critical line and the thermodynamics of
different phases of QCD which have been obtained by lattice simulations with an
imaginary chemical potential, and review motivations and merits of the
different strategies -- Taylor expansion, Fourier analysis and Pade'
approximants -- for analytic continuation from imaginary to real chemical
potential. I consider phenomenological models which can be easily extended to
the complex chemical potential plane, thus affording a direct comparison with
lattice data at imaginary $\mu$: the hadronic phase and the high temperature
limit are amenable to a simple description, while a rather subtle interplay
between thermodynamics and critical behaviour emerges in the hot phase close to
$T_c$
