  We study the phase diagram of the two-dimensional 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 Monte Carlo simulations on square
lattices $L\times L$, $L\lesssim 10^3$. We show that the low-temperature phase
of these models is controlled by the same line of Gaussian fixed points as in
the standard XY model. We find that, if a model undergoes a unique transition
by varying temperature, then the transition is of first order. In the opposite
case we observe two very close transitions: a transition associated with the
spin degrees of freedom and, as temperature increases, a transition where
chiral modes become critical. If they are continuous, they belong to the
Kosterlitz-Thouless and to the Ising universality class, respectively. Ising
and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime,
which is universal to some extent. In the chiral case, the approach is
nonmonotonic for most observables, and there is a wide region in which
finite-size scaling is controlled by an effective exponent $\nu_{\rm eff}
\approx 0.8$. This explains the result $\nu\approx 0.8$ of many previous
studies using smaller lattices.
