From 1992 to 1993, Marrow was appointed as postdoctoral research associate in the Department of Materials, University of Oxford, and a junior research fellow at Linacre College, Oxford, but moved with an Engineering and Physical Sciences Research Council (EPSRC) postdoctoral research fellowship to the School of Metallurgy and Materials, University of Birmingham. In 2001, he joined the Manchester Materials Science Centre, University of Manchester, as senior lecturer in physical metallurgy, where he became assistant director of Materials Performance Centre in 2002 and the director in 2009. 
Marrow moved to the University of Oxford to become Oxford Martin School co-director of the school programme in Nuclear and Energy Materials from 2010 to 2015, Professor in Energy Materials, Department of Materials, Oxford University, and Fellow of Mansfield College, Oxford. , Marrow is the Associate Head of Department of Materials (Teaching).
Marrow is a council member of the UK Forum for Engineering Structural Integrity (FESI), UK representative for the European Energy Research Alliance Joint Programme on Nuclear Materials, member (ex-chair) of the OECD-NEA Expert Group on Innovative Structural Materials, independent advisor to the UK Office of Nuclear Regulation on materials/structural integrity, and UK representative on Graphite for BEIS to the Generation IV International Forum. Marrow is the co-director of the Nuclear Research Centre (NRC), which is a joint venture between the University of Bristol and the University of Oxford to train new nuclear scientists and engineers.