  The basic first-order differential operators of spin geometry that are Dirac
operator and twistor operator are considered. Special types of spinors defined
from these operators such as twistor spinors and Killing spinors are discussed.
Symmetry operators of massless and massive Dirac equations are introduced and
relevant symmetry operators of twistor spinors and Killing spinors are
constructed from Killing-Yano (KY) and conformal Killing-Yano (CKY) forms in
constant curvature and Einstein manifolds. The squaring map of spinors gives KY
and CKY forms for Killing and twistor spinors respectively. They constitute a
graded Lie algebra structure in some special cases. By using the graded Lie
algebra structure of KY and CKY forms, extended Killing and conformal
superalgebras are constructed in constant curvature and Einstein manifolds.
