Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary?

I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?

Moreover, is there a theorem which classifies the 2-manifolds with boundary?

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Yes, there’s a classification theorem for non-compact 2-manifolds.

This paper gives the classification for triangulable 2-manifolds:

That an arbitrary (2nd countable, Hausdorff) topological 2-manifold admits a triangulation is fairly classical. Ahlfors book “Riemann Surfaces” has a proof. There are others available, see for example this list:

If all you’re interested in is compact manifolds with boundary, you get that classification immediately from the closed manifold case. Because if you have a compact manifold with boundary, its boundary is a disjoint union of circles. So cap those circles off with discs to produce a closed manifold. So compact manifolds with boundary are classified by the closed manifold you get by “capping off” and the number of boundary circles you started with.

Non-compact manifolds have a more delicate classification — think for example about the complement of a Cantor set in a compact surface.

Amazingly, the complete classification of noncompact 2-manifolds with boundary was not completed until 2007. Here’s the reference:

A. O. Prishlyak and K. I. Mischenko, Classification of noncompact surfaces with
boundary, Methods Funct. Anal. Topology 13 (2007), no. 1, 62–66.