Albrecht graduated in 1979 from Cornell University, Ithaca and was awarded a doctorate in 1983 at University of Pennsylvania, Philadelphia on cosmology. His thesis advisor was Paul Steinhardt.
He later carried out post-doctoral research at University of Texas, Austin and at Los Alamos National Laboratory. Albrecht later worked at Fermilab from 1987 to 1992 and subsequently taught at Imperial College, London from 1992 to 1998.
Albrecht is a Fellow of the American Physical Society and Fellow of the Institute of Physics (UK).