  We present the results of the first imaging X-ray observations in the
0.4-10.0 keV bandpass of the nearby starburst galaxies M82 and NGC 253. The
ASCA spectra of both M82 and NGC 253 are complex with strong line emission from
O, Ne, Fe, Mg, S, and Si, allowing elemental abundances to be estimated in the
X-ray band for the first time in these sources. Two components are required to
fit the spectra of both galaxies, with a "soft" component well described by a
thermal model with a temperature of ~ 10^6-7 K and a "hard" component well
described by either a thermal model (T ~ 10^8 K) or a power-law model (energy
index ~ 0.8-1.0). We find that different models (with different continua) yield
absolute abundances that differ by more than an order of magnitude, while
relative abundances are more robust and suggest an underabundance of Fe
(inferred from the Fe-L complex) relative to alpha-burning elements, possibly
as a result of dust depletion. Most of the soft flux (which originates mostly
within the central kpc of M82 and NGC 253) is consistent with starburst models
of supernovae-heated ISM and, to a lesser extent, starburst-driven superwind
emission and the direct emission from supernova (SN). The hard component in
both galaxies may have some contribution from ~ 10^8 K superwind emission or
individual SN, although most of the emission probably originates in point
sources (most likely blackhole candidates or mini-AGN with M_Edd > 3-20 solar
masses) and, possibly, inverse-Compton scattering of IR photons. The similarity
of the spectral X-ray characteristics of NGC 253 and M82 to some LINERs and
low-luminosity AGN suggests a link between AGN and starbursts (e.g., both may
contain an accretion-driven emission component).
