Sub-Saharan Africa is among the regions that contribute least to global climate change, yet it is among the most vulnerable to its impacts due to low levels of economic and technological development. The frequency and magnitude of extreme weather events in the region are rising at a faster rate than the population capacity to deal with the attendant disasters. This paper interrogates some emerging and existing evidence of the potential for extreme weather events to obviate countries' attainment of the Sustainable Development Goals (SDGs). Whilst previous studies have assessed the vulnerabilities of sub-Saharan African (SSA) countries to extreme weather events on specific sectors, a comprehensive assessment of the implications of these extreme events for attaining the SDGs remains largely untouched. This paper assesses the impacts of flooding, extreme heat and drought on five key SDGs—Zero hunger (Goal 2), Good health and well-being (Goal 3), Quality education (Goal 4), Clean water and sanitation (Goal 6), and Sustainable cities and communities (Goal 11). Based on empirical cases from different SSA country contexts, and guided by the SDG targets and indicators, we discuss the main interactions between extreme weather events and different SDGs, emerging with a framework for the climate change—sustainable development nexus. Such an assessment, with regard to specific national and local case studies, would inform policy formulation and implementation, research and investment toward sustainable development in the region. Integrating resilence strategies into national development policies will offer sub-Saharan African countries the opportunity to reduce the impacts of extreme weather events on attaining their targets for sustainable development towards Agenda 2030.