  In this paper we study exactly marginal deformations of field theories living
on D3-branes at low energies. These theories include N=4 supersymmetric
Yang-Mills theory and theories obtained from it via the orbifolding procedure.
We restrict ourselves only to orbifolds and deformations which leave some
supersymmetry unbroken. A number of new families of N=1 superconformal field
theories are found. We analyze the deformations perturbatively, and also by
using general arguments for the dimension of the space of exactly marginal
deformations. We find some cases where the space of perturbative exactly
marginal deformations is smaller than the prediction of the general analysis at
least up to three-loop order), and other cases where the perturbative result
(at low orders) has a non-generic form.
