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 \sim 10 ^ { 6 - 7 } K and a “ hard ” component well described by either a thermal model ( T _ { hard } \sim 10 ^ { 8 } K ) or a power-law model ( \Gamma \sim 1.8 - 2.0 ) . 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 . Dust depletion may be responsible for the low relative abundance of Fe inferred from the soft-component fits . We set the most reliable limits to date on the Fe-K emission line equivalent width ( EW ) , with an upper limit in NGC 253 of EW < 180 eV while a line at \sim 6.6 keV is marginally significant ( at the 3 \sigma level ) with EW \sim 100 eV in M82 . The low Fe-K line emission EW limits suggest either significantly sub-solar abundances in the material producing the hard component ( if thermal ) or that there is a significant non-thermal or non-equilibrium contribution to the hard component . 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 \sim 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 M _ { \odot } ) 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 ) .