  We study admissible and equivalence point transformations between generalized
multidimensional nonlinear Schr\"odinger equations and classify Lie symmetries
of such equations. We begin with a wide superclass of Schr\"odinger-type
equations, which includes all the other classes considered in the paper.
Showing that this superclass is not normalized, we partition it into two
disjoint normalized subclasses, which are not related by point transformations.
Further constraining the arbitrary elements of the superclass, we construct a
hierarchy of normalized classes of Schr\"odinger-type equations. This gives us
an appropriate normalized superclass for the non-normalized class of
multidimensional nonlinear Schr\"odinger equations with potentials and modular
nonlinearities and allows us to partition the latter class into three families
of normalized subclasses. After a preliminary study of Lie symmetries of
nonlinear Schr\"odinger equations with potentials and modular nonlinearities
for an arbitrary space dimension, we exhaustively solve the group
classification problem for such equations in space dimension two.
