We introduce notions of finiteness obstruction, Euler characteristic,
L^2-Euler characteristic, and M\"obius inversion for wide classes of
categories. The finiteness obstruction of a category \Gamma of type (FP) is a
class in the projective class group K_0(R\Gamma); the Euler characteristic and
L^2-Euler characteristic are respectively its R\Gamma-rank and L^2-rank. We
also extend the second author's K-theoretic M\"obius inversion from finite
categories to quasi-finite categories. Our main example is the proper orbit
category, for which these invariants are established notions in the geometry
and topology of classifying spaces for proper group actions. Baez-Dolan's
groupoid cardinality and Leinster's Euler characteristic are special cases of
the L^2-Euler characteristic. Some of Leinster's results on M\"obius-Rota
inversion are special cases of the K-theoretic M\"obius inversion.