  Two-dimensional $CP^{N-1}$ models are investigated by Monte Carlo methods on
the lattice, for values of $N$ ranging from 2 to 21. Scaling and rotation
invariance are studied by comparing different definitions of correlation length
$\xi$. Several lattice formulations are compared and shown to enjoy scaling for
$\xi$ as small as $2.5$. Asymptotic scaling is investigated using as bare
coupling constant both the usual $\beta$ and $\beta_E$ (related to the internal
energy); the latter is shown to improve asymptotic scaling properties. Studies
of finite size effects show their $N$-dependence to be highly non-trivial, due
to the increasing radius of the $\bar z z$ bound states at large $N$.
