Joseph Curran completed his doctoral studies in the Department of Biochemistry at Queens University Belfast, where he was awarded a PhD in 1984. He then joined the UNIGE Faculty of Medicine as a post-doctoral fellow in the Department of Microbiology and Molecular Medicine, where he worked on transcriptional and translational regulation in the parainfluenza virus. He was appointed Senior Lecturer in 1993, and in 2002 he was promoted to Privat-Docent, continuing his work in virology.

From 2006 onwards, thanks to support from the SNSF and the Geneva Cancer League, he focused his research on the implications of dysfunctional translational regulation for cell proliferation. His current work focuses on the heterogeneity observed in the non-coding regions of cellular mRNA transcripts. In 2013, Joseph Curran was promoted to Associate Professor in the Department of Microbiology and Molecular Medicine. Since then, he has held the position of Biosatety Officer (BSO) for the CMU and has set up a new system for monitoring notifications via the ECOGEN website.

Very involved in pre-graduate teaching, Joseph Curran taught all three years of the Bachelor's programme in human medicine.  He also teaches virology to students in dental medicine. He is appointed Emeritus Professor in October 2025.