Weaver was born in Victoria, British Columbia, and graduated from Oak Bay High School in 1979.
He received a B.Sc in mathematics and physics from the University of Victoria in 1983, a Certificate of Advanced Studies in Mathematics (Master of Advanced Study) from Cambridge University in 1984, and a PhD in Applied Mathematics from the University of British Columbia in 1987.
After finishing his PhD, Weaver worked as a postdoctoral fellow in the School of Mathematics at the University of New South Wales in Australia in 1988, and in the Joint Institute for the Study of the Atmosphere and Ocean at the University of Washington in Seattle in 1989. Prior to joining the University of Victoria in 1992, he spent three years as an assistant professor in the department of Atmospheric and Oceanic Sciences at McGill University. Weaver is a Lansdowne Professor and, prior to his election to the BC Legislature, was the Canada Research Chair in climate modelling and analysis in the School of Earth and Ocean Sciences at the University of Victoria, where he has worked for 25 years.
Weaver has chaired or served as a member of numerous local, national and international committees. From 2003-2004 he was president of the Victoria Confederation of Parent Advisory Councils (VCPAC). In 2004-2005 he was president of the University of Victoria Faculty Association and served as their chief negotiator in the 2003 and 2006 collective bargaining. Weaver has been engaged in public outreach and science communication. He sat on the CRD Roundtable on the Environment, and has delivered numerous public and school presentations and hosted many school field trips to his university laboratory over his career. He continues to lead the development of the Vancouver Island School Based weather station project.
Weaver has authored or coauthored over 200 peer-reviewed papers in climate, meteorology, oceanography, earth science, policy, education and anthropology journals. He was a lead author in the United Nations Intergovernmental Panel on Climate Change 2nd, 3rd, 4th and 5th scientific assessments. He was the chief editor of the Journal of Climate from 2005-2009.
Weaver is a fellow of the Royal Society of Canada, Canadian Meteorological and Oceanographic Society, the American Geophysical Union, the American Meteorological Society and the American Association for the Advancement of Science.
Over the years he has received numerous awards including the E.W.R. NSERC Steacie Fellowship in 1997, the Killam Research Fellowship and a CIAR Young Explorers award as one of the top 20 scientists in Canada under the age of 40 in 2002, the CMOS President's Prize in 2007, a Guggenheim Fellowship in 2008 and the Royal Society of Canada Miroslaw Romanowski Medal and the A.G. Huntsman Award for Excellence in Marine Science in 2011. In 2008 he was appointed to the Order of British Columbia and in 2013 he was awarded a Queen's Diamond Jubilee Medal. In 2014, Weaver received an Honorary DSc from McMaster University.
His book, Keeping our Cool: Canada in a Warming World was published by Viking Canada in September 2008 (ISBN 978-0-670-06800-5). His second book, Generation Us: The Challenge of Global Warming was published by Raven Books in 2011 (ISBN 978-1-55469-804-2).
Weaver joined the British Columbia Green Party in October 2012 as the party's deputy leader and candidate for the riding of Oak Bay-Gordon Head. He was elected in the 2013 provincial election as the first Green Party MLA in British Columbia's history.
Since his running for office, Weaver has been a strong supporter of the development of the clean technology industry and a firm critic of the Liberal government's liquefied natural gas strategy. Weaver calls the LNG plan a "pipe-dream", asserting that Liberal government promises of a $1 trillion boost to the provincial GDP, a $100 billion prosperity fund, and the elimination of the provincial debt and sales tax are unsubstantiated and irresponsible. Weaver has also been involved in several local issues, most recently calling for a reconsideration of the Capital Regional District's sewage treatment plan in 2013.
In August 2013, Weaver chose not to take on leadership of the B.C. Green Party stating: "I have an ambitious agenda for my term as MLA and achieving this requires focus and hard work. I consider it in the best interests of my constituents, the party and the province if, for now, I focus on my role as MLA and support a new interim leader who can concentrate on building the party."
However, Weaver also stated that if he decides to re-run in the 2017 provincial election and is still the only elected B.C. Green Party MLA, he would then seek leadership of the party.
On 24 November 2015 Weaver announced his bid to run for leader of the British Columbia Green Party. He was acclaimed to this position on December 9, 2015.
The British Columbia Supreme Court ruled that Dr. Weaver had been libelled in a series of National Post articles written by Terence Corcoran, Peter Foster and Kevin Libin that accused him of scientific misconduct in his studies of climate change, among other claims. In Weaver v. Corcoran the court ruled that the charges were false and that "the defamation in this case was serious. It offended Dr. Weaver's character and the defendants refused to publish a retraction". The court awarded Weaver $50,000 in damages, ordered that the offending articles be removed from the newspaper's archives and that a complete retraction be published. The National Post appealed the decision. On April 21, 2017, The B.C. Court of Appeal ordered a retrial in the defamation suit, citing an error in the judge's analysis of the articles pertaining to the suit.
Weaver is featured in the documentary film Running on Climate. Filming began before Weaver was nominated as a candidate and continued through the 2013 election campaign.