The broadband X-ray spectrum of the prototypical starburst galaxg M82 is very complex. At least three spectral components are required to fit the combined ROSAT and ASCA spectrum in the 0.1-10 keV range. The observed X-ray flux in this band is dominated by a hard Gamma=1.7, heavily absorbed power-law component which originates in the nucleus and near-nuclear disk of the galaxy. Among the candidates for the origin of this hard X-ray emission, the most plausible appears to be inverse-Compton scattered emission from the interaction of the copious infrared photon flux of M82 with supernova-generated relativistic electrons. The measured intrinsic luminosity of the power-law component agrees closely with calculations of the expected inverse-Compton luminosity, Moreover, the radio and X-ray emission in the nucleus of M82 have the same spectral slope, which should be the case if both types of emission are nonthermal and ape associated with a common population of electrons. The other two spectral components, thermal plasmas with characteristic temperatures kT approximate to 0.6 and 0.3 keV, are associated with the star formation and starburst-driven wind in M82. The warmer thermal component is heavily absorbed as well and must also originate in the central region of the galaxy. The softer thermal component, however, is not absorbed, and is likely to represent the X-ray emission that ex;ends along tie minor axis of M82. The amount of absorption required in the three-component model suggests that the intrinsic luminosity of M82 in the 0.1-10 keV band is about four times greater than its observed luminosity of 4 x 10(40) ergs s(-1).