TITLE: CW complexes and manifolds
QUESTION [34 upvotes]: What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes?
Thank you guys. This is just  a question I thought of during class, obviously not homework...

REPLY [22 votes]: This is an addendum to Neal's answer (concerning the 4-dimensional case):

Every connected noncompact 4-manifold admits a PL structure, hence, a triangulation (F.Quinn, "Ends of maps-III", 1982). 
For compact 4-manifolds the problem of existence of a CW-complex (as well as a handlebody) structure is open, see Question 1.3 in this list of problems.