A Lagrangian definition of a large family of (0,2) supersymmetric conformal
field theories may be made by an appropriate gauge invariant combination of a
gauged Wess-Zumino-Witten model, right-moving supersymmetry fermions, and
left-moving current algebra fermions. Throughout this paper, use is made of the
interplay between field theoretic and algebraic techniques (together with
supersymmetry) which is facilitated by such a definition. These heterotic coset
models are thus studied in some detail, with particular attention paid to the
(0,2) analogue of the N=2 minimal models, which coincide with the `monopole'
theory of Giddings, Polchinski and Strominger. A family of modular invariant
partition functions for these (0,2) minimal models is presented. Some examples
of N=1 supersymmetric four dimensional string theories with gauge groups E_6 X
G and SO(10) X G are presented, using these minimal models as building blocks.
The factor G represents various enhanced symmetry groups made up of products of
SU(2) and U(1).