  We study the phase diagram and critical behavior of the two-dimensional
square-lattice fully frustrated XY model (FFXY) and of two related models, a
lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the
critical modes of the FFXY model, and a coupled Ising-XY model. We present a
finite-size-scaling analysis of the results of high-precision Monte Carlo
simulations on square lattices L x L, up to L=O(10^3).
  In the FFXY model and in the other models, when the transitions are
continuous, there are two very close but separate transitions. There is an
Ising chiral transition characterized by the onset of chiral long-range order
while spins remain paramagnetic. Then, as temperature decreases, the systems
undergo a Kosterlitz-Thouless spin transition to a phase with quasi-long-range
order.
  The FFXY model and the other models in a rather large parameter region show a
crossover behavior at the chiral and spin transitions that is universal to some
extent. We conjecture that this universal behavior is due to a multicritical
point. The numerical data suggest that the relevant multicritical point is a
zero-temperature transition. A possible candidate is the O(4) point that
controls the low-temperature behavior of the 4-vector model.
