29 pages.We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \alpha in terms of the ranks of certain groups which appear in the construction. We give bounds for \alpha. These computations apply to some well known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over \Q. We show that such a link cannot be established for more general tilings, and we present a counter-example in dimension one
