The geometry of a smooth projective variety is encoded in its derived category of coherent sheaves, which is a smooth compact triangulated category. Such categories can then be considered as categories of coherent sheaves on smooth compact "noncommutative" manifolds (which may or may not exist as geometric objects). I will discuss two possible notions of dimension for a smooth compact triangulated category: the Serre dimension and the Rouquier dimension. This will be mostly expository talk, which I will try to make accessible to non-experts and graduate students. (Based on joint work in progress with Alexey Elagin).