We introduce and study a new class of projection methods -- namely the
velocity-correction methods in standard form and in rotational form --
for solving the unsteady incompressible Navier--Stokes equations.  We
show that the rotational form provides improved error estimates in
terms of the $H^1$-norm for the velocity and of the $L^2$-norm for the
pressure.  We also show that the class of fractional-step methods
introduced in \cite{OID} and \cite{KIO} can be interpreted as the
rotational form of our velocity-correction methods. Thus, to the best
of our knowledge, our results provide the first rigorous proof of
stability and convergence of the methods in \cite{OID} and \cite{KIO}.
We also emphasize that contrary to those of \cite{OID} and \cite{KIO},
our formulations are set in the standard $L^2$ setting, and
consequently can be easily implemented by means of any variational
approximation techniques, in particular the finite element methods.