Page "Riemann surface" Paragraph 3

Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

A two-dimensional real manifold can be turned into a Riemann surface ( usually in several inequivalent ways ) if and only if it is orientable and metrizable.

So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.