  This article reviews the current status of precursor superconducting phase
fluctuations as a possible mechanism for pseudogap formation in
high-temperature superconductors. In particular we compare this approach which
relies on the two-dimensional nature of the superconductivity to the often used
$T$-matrix approach. Starting from simple pairing Hamiltonians we present a
broad pedagogical introduction to the BCS-Bose crossover problem. The finite
temperature extension of these models naturally leads to a discussion of the
Berezinskii-Kosterlitz-Thouless superconducting transition and the related
phase diagram including the effects of quantum phase fluctuations and
impurities. We stress the differences between simple Bose-BCS crossover
theories and the current approach where one can have a large pseudogap region
even at high carrier density where the Fermi surface is well-defined. The
Green's function and its associated spectral function, which explicitly show
non-Fermi liquid behaviour, is constructed in the presence of vortices. Finally
different mechanisms including quasi-particle-vortex and vortex-vortex
interactions for the filling of the gap above $T_c$ are considered.
