Burghelea was born in Râmnicu Vâlcea, Romania, in 1943, where he attended Alexandru Lahovari National College (at that time lyceum Nicolae Bălcescu).

He obtained his Ph.D. in 1968 from the Institute of Mathematics of the Romanian Academy (IMAR) with a thesis on Hilbert manifolds.

[2] In 1972, Burghelea was awarded the title of Doctor Docent in sciences by the University of Bucharest, making him the youngest recipient of the highest academic degree in Romania.

[6] His most significant contributions are on Topology of infinite dimensional manifolds;[7][8] Homotopy type of the space of homeomorphisms and diffeomorphisms of compact smooth manifolds;[9][10] Algebraic K-theory and cyclic homology of topological spaces, groups (including simplicial groups) and commutative algebras (including differential graded commutative algebras);[11][12][13] Zeta-regularized determinants of elliptic operators and implications to torsion invariants for Riemannian manifolds.

He has authored several books including Groups of Automorphisms of Manifolds and New Topological Invariants for Real- and Angle-valued Maps: An Alternative to Morse-Novikov Theory.