  After a brief historical survey that emphasizes the role of the algebra
obeyed by the Dirac operator, we examine an algebraic Dirac operator associated
with Lie algebras and Lie algebra cosets. For symmetric cosets, its
``massless'' solutions display non-relativistic supersymmetry, and can be
identified with the massless degrees of freedom of some supersymmetric
theories: N=1 supergravity in eleven dimensions (M-theory), type IIB string
theory in ten and four dimensions, and in four dimensions, N=8 supergravity,
N=4 super-Yang-Mills, and the N=1 Wess-Zumino multiplet. By generalizing this
Dirac operator to the affine case, we generate superconformal algebras
associated with cosets ${\bf g}/\bf h$, where $\bf h$ contains the {\it space}
little group. Only for eleven dimensional supergravity is $\bf h$ simple. This
suggests, albeit in a non-relativistic setting, that these may be the limit of
theories with underlying two-dimensional superconformal structure.
