We present models that predict spectra of old- and intermediate-aged stellar populations at 2.51 Å (FWHM) with varying [α/Fe] abundance. The models are based on the MILES library and on corrections from theoretical stellar spectra. The models employ recent [Mg/Fe] determinations for the MILES stars and BaSTI scaled-solar and α-enhanced isochrones. We compute models for a suite of initial mass function (IMF) shapes and slopes, covering a wide age/metallicity range. Using BaSTI, we also compute `base models' matching the Galactic abundance pattern. We confirm that the α-enhanced models show a flux excess with respect to the scaled-solar models blueward ˜4500 Å, which increases with age and metallicity. We also confirm that both [MgFe] and [MgFe]' indices are [α/Fe]-insensitive. We show that the sensitivity of the higher order Balmer lines to [α/Fe] resides in their pseudo-continua, with narrower index definitions yielding lower sensitivity. We confirm that the α-enhanced models yield bluer (redder) colours in the blue (red) spectral range. To match optical colours of massive galaxies, we require both α-enhancement and a bottom-heavy IMF. The comparison of globular cluster line-strengths with our predictions match the [Mg/Fe] determinations from their individual stars. We obtain good fits to both full spectra and indices of galaxies with varying [α/Fe]. Using thousands of SDSS galaxy spectra, we obtain a linear relation between a proxy for the abundance, [ZMg/ZFe]SS(BaSTI), using solely scaled-solar models and the [Mg/Fe] derived with models with varying abundance ([Mg/Fe] = 0.59[ZMg/ZFe]SS(BaSTI)). Finally, we provide a user-friendly, web-based facility, which allows composite populations with varying IMF and [α/Fe].