Welsh was born in Regina, Saskatchewan. She received her Bachelor of Arts degree from the University of Saskatchewan and then went on to become a Rhodes Scholar earning her Masters and Doctorate in International Relations from the University of Oxford.
She currently resides in Montreal, where she is the Canada 150 Research Chair in Global Governance and Security and serves as the Director of the Centre for International Peace and Security Studies (CIPSS) at McGill University. She was previously Professor and Chair in International Relations at the European University Institute. She is a senior research fellow at Somerville College, University of Oxford. Welsh was also a professor of International Relations at the University of Oxford and co-director of the Oxford Institute for Ethics, Law and Armed Conflict at the James Martin 21st Century School. She has also served as consultant to a number of organizations including the Conference Board of Canada, McKinsey and Co, the Aspen Institute's Business and Society Program and the Government of Canada.
Welsh is the author of two books and co-editor of several others. Her latest book is At Home in the World: Canada's Global Vision for the 21st Century. In it, she provides an alternative vision for Canada's role in international relations and contends that Canada is a "Model Citizen" not a "Middle Power". Welsh suggests that Canada can play a major role in International policy if it provides a model for other countries to follow.
Welsh delivered the 2016 CBC Massey Lectures, entitled The Return of History.