Finite field theory on noncommutative geometries

The propagator is calculated on a noncommutative version of the flat plane and the Lobachevsky plane with and without an extra (Euclidean) time parame ter. In agreement with the general idea of noncommutative geometry it is fo und that the limit; when the two "points" coincide is finite and diverges o nly when the geometry becomes commutative. The flat four-dimensional case i s also considered. This is at the moment less interesting since there has b een no curved case developed with which it can be compared.