a triangulation of a topological space is a homeomorphism from this space to a simplicial complex. a famous problem in topology is whether all manifolds are triangulable. surprisingly, the answer is no when dimension is at least 4. in this talk, i will explain the beautiful work of galewski-stern and matumoto, which provides an obstruction theory for triangulating manifolds. i will also explain manolescu's disproof of triangulation conjecture in all dimensions greater than 4.