We introduce the periodic Airy-Schrödinger operator and we study its band spectrum. This is an example of an explicitely solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a " classical regime " , a " semiclassical regime " and a " semiclassical limit ". We prove that there exists an explicit constant, which is a zero of a classical function, which marks the transition between the classical and semiclassical regimes. We completely determine the behaviour of the edges of the first spectral band with respect to our semiclassical parameter. Then, we investigate the spectral bands situated in the range of the potential. In the semiclassical regime, we prove precise estimates on the widths of the spectral bands and the spectral gaps and we determine an upper bound on the spectral density in this range. Finally, in the semiclassical limit, we get asymptotics expansions of the edges of the spectral bands and thus of the widths of the spectral bands and the spectral gaps.