  We fit the surface-brightness profiles of 21 elliptical galaxies using both
the Sersic function and a new empirical model which combines an inner power law
with an outer Sersic function. The profiles are combinations of deconvolved HST
profiles from the literature and ellipse fits to the full WFPC2 mosaic images,
and thus span a radial range from ~ 0.02 arcsec to ~ twice the half-light
radius. We are able to accurately fit the entire profiles using either the
Sersic function or our new model. In doing so, we demonstrate that most, if not
all, so-called ``power-law'' galaxies are better described as ``Sersic
galaxies'' -- they are well modeled by the three-parameter Sersic profile into
the limits of HST resolution -- and that ``core'' galaxies are best understood
as consisting of an outer Sersic profile with an inner power-law cusp, which is
a downward deviation from the inward extrapolation of the Sersic profile. This
definition of cores resolves ambiguities that result when the popular ``Nuker
law'' is fitted to the profiles of ellipticals and bulges, particularly at
lower luminosities. We also find that using the Nuker law to model core-galaxy
nuclear profiles systematically overestimates the core radii by factors of
1.5--4.5 and underestimates the inner power-law slope by ~ 20--40% or more.
