  We prove a version of the classical Runge and Mergelyan uniform approximation
theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we
obtain some geometric applications. Among them, we emphasize the following
ones:
  1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces.
  2. An existence theorem for non-orientable minimal surfaces in R3, with
arbitrary conformal structure, properly projecting into a plane.
  3. An existence result for non-orientable minimal surfaces in R3 with
arbitrary conformal structure and Gauss map omitting one projective direction.
