We present two new NuSTAR observations of the narrow-line Seyfert 1 (NLS1) galaxy Mrk 766 and give constraints on the two scenarios previously proposed to explain its spectrum and that of other NLS ls: relativistic reflection and partial covering. The NuSTAR spectra show a strong hard (>15 keV) X-ray excess, while simultaneous soft X-ray coverage of one of the observations provided by XMM-Newton constrains the ionized absorption in the source. The pure reflection model requires a black hole of high spin (a > 0.92) viewed at a moderate inclination (i = 46(-4)(-1)degrees). The pure partial covering model requires extreme parameters: the cut-off of the primary continuum is very low (22(-5)(+7) keV) in one observation and the intrinsic X-ray emission must provide a large fraction (75 per cent) of the bolometric luminosity. Allowing a hybrid model with both partial covering and reflection provides more reasonable absorption parameters and relaxes the constraints on reflection parameters. The fractional variability reduces around the iron K band and at high energies including the Compton hump, suggesting that the reflected emission is less variable than the continuum.