  We investigate the structure of Parisi measures, the functional order
parameters of mixed p-spin models in mean field spin glasses. In the absence of
external field, we prove that a Parisi measure satisfies the following
properties. First, at all temperatures, the support of any Parisi measure
contains the origin. If it contains an open interval, then the measure has a
smooth density on this interval. Next, we give a criterion on temperature
parameters for which a Parisi measure is neither Replica Symmetric nor One
Replica Symmetry Breaking. Finally, we show that in the Sherrington-Kirkpatrick
model, slightly above the critical temperature, the largest number in the
support of a Parisi measure is a jump discontinuity. An analogue of these
results is discussed in the spherical mixed p-spin models. As a tool to
establish these facts and of independent interest, we study functionals of the
associated Parisi PDEs and derive regularity properties of their solutions.
