TITLE: What is the easiest way to classify all possible smooth orientable closed 2-manifolds?
QUESTION [6 upvotes]: If this has been answered already, please let me know and I'll delete the question. 
ADDED: I'd prefer to assume a smooth structure, rather than a triangulation.

REPLY [5 votes]: A very simple and low tech solution to this problem is the very first proof, 
given by Möbius in 1863. He assumes that the surface is smoothly embedded
in $\mathbb{R}^3$, and slices it by a family of parallel planes. Assuming that
the orientation of the planes is general and that they are sufficiently close together, this cuts the surface into simple pieces -- either disks, annuli, or pairs of pants.
It is then quite easy to show that the result of assembling such pieces is always a sphere with handles.