Which manifolds are homeomorphic to simplicial complexes? This question is only motivated by curiosity; I don't know a lot about manifold topology.
Suppose $M$ is a compact topological manifold of dimension $n$.  I'll assume $n$ is large, say $n\geq 4$.  The question is: Does there exist a simplicial complex which is homeomorphic to $M$?
What I think I know is:

*

*If $M$ has a piecewise linear (PL) structure, then it is triangulable, i.e., homeomorphic to a simplicial complex.


*There is a well-developed technology ("Kirby-Siebenmann invariant") which tells you whether or not a topological manifold admits a PL-structure.


*There are exotic triangulations of manifolds which don't come from a PL structure.  I think the usual example of this is to take a homology sphere $S$ (a manifold with the homology of a sphere, but not maybe not homeomorphic to a sphere), triangulate it, then suspend it a bunch of times.  The resulting space $M$ is supposed to be homeomorphic to a sphere (so is a manifold).  It visibly comes equipped with a triangulation coming from that of $S$, but has simplices whose link is not homemorphic to a sphere; so this triangulation can't come from a PL structure on $M$.
This leaves open the possibility that there are topological manifolds which do not admit any PL-structure but are still homeomorphic to some simplicial complex.  Is this  possible?
In other words, what's the difference (if any) between "triangulable" and "admits a PL structure"?
This Wikipedia page on 4-manifolds claims that the E8-manifold is a topological manifold which is not homeomorphic to any simplicial complex; but the only evidence given is the fact that its Kirby-Siebenmann invariant is non trivial, i.e., it doesn't admit a PL structure.
 A: Regarding Charles Rezk's second question:
This leaves open the possibility that there are topological manifolds which do not admit any PL-structure but are still homeomorphic to some simplicial complex. Is this possible?
For dimension 4, it follows from the Poincare conjecture that a 4-manifold is triangulable iff smoothable (which is also equivalent to having PL structure for dimension <8). See Problem 3 of Fragments of geometric topology from the sixties by Sandro Buoncristiano. Also see the presentation From Triangulations to 4-Manifolds: In Honor of Takao Matumoto’s 60th Birthday by Ron Stern.
For dimension >4, Springer Online Reference Works claims that "the imbedding $PL \subset TRI$ is also irreversible in the same strong sense (there exist polyhedral manifolds of dimension $\geq 5$ that are homotopy inequivalent to any PL-manifold)",  but gives no examples. In Ron Stern's presentation it is stated that "All oriented closed 5-manifolds triangulable", so I think among them there may be some with nontrivial KS invariant and hence cannot bear PL structure.
In addition, the book Lectures on the Topology of 3-manifolds: An Introduction to the Casson Invariant (p.168, Theorem 18.4) by Nikolai Saveliev seems to contain a result that strengthens the one mentioned in Paul's answer.
Added: The paper Piecewise linear structures on topological manifolds (22.5. Example) by Yuli B. Rudyak explicitly gives an example of "A topological manifold which is homeomorphic to a polyhedron but does not admit any PL structure".
A: For a discussion of the 4-dimensional case see 
http://www.map.mpim-bonn.mpg.de/Questions_about_surgery_theory.
A: Galewski-Stern proved
https://mathscinet.ams.org/mathscinet-getitem?mr=420637
" It follows that every topological m-manifold, m≥7 (or m≥6 if ∂M=∅), can be triangulated if and only if there exists a PL homology 3-sphere H3 with Rohlin invariant one such that H3#H3 bounds a PL acyclic 4-manifold."
The Rohlin invariant is a Z/2 valued homomorphsim on the 3-dimensional homology cobordism group, $\Theta_3\to Z/2$, so if it splits there exist non-triangulable manifodls in high dimensions.
A: I don't know about dimension 4, but for high dimensions this is a well-known open problem.  I don't think much progress has been made on it for a while.  I recommend Ranicki's lecture notes from Siebenmann's retirement conference for a good summary about what is known about this and related problems:
https://www.maths.ed.ac.uk/~v1ranick/slides/orsay.pdf
EDIT : Hot off the press is a paper of Manolescu claiming to disprove the conjecture of Galewski-Stern and construct manifolds in all dimensions $\geq 5$ which are not homeomorphic to simplicial complexes.
