Supernova remnants may exhibit both thermal and nonthermal X-ray emission. Such remnants can be distinguished by the weakness of their X-ray lines because of the presence of a strong nonthermal X-ray continuum. RCW 86 is a remnant with weak lines, resulting in low and peculiar abundances when thermal models alone are used to interpret its X-ray spectrum. This indicates the presence of a strong nonthermal synchrotron continuum. We analyze ASCA X-ray spectra of RCW 86 with the help of both nonequilibrium ionization thermal models and nonthermal synchrotron models. A two-temperature thermal model and a simple nonthermal model with an exponential cutoff (plus interstellar absorption) give reasonable results. We obtain a blast-wave velocity of 800 km s-1, a shock ionization age of 1×1011-3×1011 cm-3 s, and the break in nonthermal spectra at 2×1016-4×1016 Hz. The strength of the nonthermal continuum correlates well with the radio brightness in the bright southwest section of the remnant. This is convincing evidence for X-ray synchrotron emission in RCW 86.