Triangulating surfaces I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated.  I know two sources for this, namely Ahlfors's book on Riemann surfaces and Moise's book "Geometric topology in dimensions 2 and 3".  Both of these strike me as being a bit much for a bright undergraduate.  Question : in the 30+ years since Moise's book, has anyone written a more accessible account?
 A: If you're okay going the extra step and assuming a smooth structure, the standard argument of Whitehead goes like this: take a smooth embedding of your manifold (of any dimension) into euclidean space.  Triangulate Euclidean space, perturb the embedding to make it transverse to the skeleta of the triangulation.  Refine the triangulation (barycentric subdivision) to the point where the embedding "looks linear" in each top-dimensional simplex.  The triangulations of the simplices pulls-back to a polyhedral decomposition of the manifold, which you can subdivide to be a triangulation. 
If you insist on going the extra step to topological manifolds you could smooth the topological structure.  I believe much of that argument appears in Thurston's 3-dimensional geometry and topology book but I don't have it at home at the moment, and I don't remember. 
A: There is a forthcoming undergraduate differential geometry book by Christian Baer (at the University of Potsdam, Germany) which is very nice; it gives a careful account of this theorem and a very accessible proof. If the book is not publically available, it should be soon.
A: [Three years later …]
All the published proofs of triangulability of surfaces that I am aware of use the Schoenflies theorem, which is not exactly an easy thing to prove. There is however another line of proof which avoids the Schoenflies theorem and instead uses the Kirby torus trick that underlies Kirby-Siebenmann theory in higher dimensions. There is a 1974 paper by A.J.S.Hamilton that gives much simpler proofs of Moise's theorems on triangulability of 3-manifolds using the torus trick, and the same ideas can be applied even more simply for surfaces.  Instead of the Schoenflies theorem one needs a few results about surfaces strictly in the PL (or smooth if one prefers) category. Namely, one needs to know that PL structures are unique up to PL homeomorphism in the following four cases:  $S^1\times S^1$, $S^1\times{\mathbb R}$, $[0,1]\times{\mathbb R}$, and $D^2$. These can be regarded as special cases of the usual classification theorem for compact PL surfaces, extended to include a few noncompact cases.
I haven't seen this proof in the literature, so I've written it up as a short expository paper "The Kirby torus trick for surfaces" and posted it on the arXiv here, working in the smooth category rather than the PL category.
It's not clear how suitable this proof would be for an undergraduate course. Besides the ingredients mentioned above, a little basic covering space theory is also needed. If one were in the fortunate position of already having covered these things, then this proof might be accessible to undergraduates. On the other hand, it could be of some interest to go through a proof of the often-quoted-but-seldom-proved Schoenflies theorem. (In this connection I might mention a paper by Larry Siebenmann on the Schoenflies theorem in the Russian Math Surveys in 2005, giving history as well as a proof.)
A: What about the 1968 three-pages paper by Doyle and Moran?
http://link.springer.com/article/10.1007%2FBF01425546
A: This is not an answer that points to a more recent and more accessible account of the triangulability of surfaces, but rather a way to make the account in the first chapter of Ahlfors' & Sario's book more accessible, if sufficient time is available. It should be noted that the proof given by Ahlfors & Sario works for all (connected, 2nd countable) surfaces: compact or noncompact; with or without boundary. I will describe what I found to be the difficulties with Ahlfors' & Sario's presentation and how one can mitigate these difficulties, especially if the learning of this material is by self-study, as was the case for me, and not in the context of a university course. Disclaimer: I am a mathematician but not a topologist.
I found there were three main difficulties, all stemming from Ahlfors' & Sario's terse style of writing. The first difficulty is the absence of any references for background material. I found that the classic, self-contained book, Elements of the Topology of Plane Sets of Points (2nd ed.), by M.H.A. Newman, and the first two sections of the third chapter of the book, Algebraic Topology, by E. Spanier (for the basics of the theory of abstract simplicial complexes), provide sufficient background. The second difficulty is that almost every sentence resembles the statement of a lemma whose proof is left to the reader. The third difficulty is an intentionally omitted proof of a rather difficult result, "46C".
Regarding the second and third difficulties: after filling in the missing details, I decided to write them up in the form of a list of notes (as opposed to an article). I then created a website, on which I posted these notes. Included in them, is a proof of the result "46C" that relies heavily on the material in the cited book by Newman. Although I did not strive for either optimum mathematical efficiency or elegance, perhaps my notes will be useful to others, in making Ahlfors' & Sario's account more accessible.
While I was at it, I also posted some details for a proof of Schoenflies' Theorem via Complex Analysis; these are details for the presentation in the book, Boundary Behaviour of Conformal Maps, by C. Pommerenke. Note that this proof assumes the Riemann Mapping Theorem, proofs of which are more widely available. (These notes were written before I was aware of Newman's book, which happens to also contain a proof of Schoenflies' Theorem -- a proof that is purely topological in nature. As mentioned in Allen Hatcher's answer, an historical account of proofs of Schoenflies' Theorem, including a new one at the time, appeared in the paper and its errata. A preprint of the paper is freely available here.)
A careful accounting of all the prerequisite material for Ahlfors' & Sario's approach, reveals that it is a very substantial amount. Although such a proof of the triangulability of surfaces can be made accessible to undergraduates, it is not clear whether an undergraduate who embarks on such a project of self-study will still be an undergraduate upon completion of the project.
A: Try this book by Jean Gallier and Diana Xu. It is aimed at undergraduates and has a nice account of Thomassen's elementary proof of the triangulation theorem in the last appendix. Or you can refer the students to Thomassen's original paper which is also quite readable.
A: The notes (in French) for Gramain's 1969-70 course, available at
http://www.math.u-psud.fr/~biblio/numerisation/docs/G_GRAMAIN-55/pdf/G_GRAMAIN-55.pdf
seem to include a proof using Morse theory, based on a quick glance I made.
