Extreme event attribution aims to elucidate the link between global climate change, extreme weather events, and the harms experienced on the ground by people, property, and nature. It therefore allows the disentangling of different drivers of extreme weather from human-induced climate change and hence provides valuable information to adapt to climate change and to assess loss and damage. However, providing such assessments systematically is currently out of reach. This is due to limitations in attribution science, including the capacity for studying different types of events, as well as the geographical heterogeneity of both climate and impact data availability. Here, we review current knowledge of the influences of climate change on 5 different extreme weather hazards (extreme temperatures, heavy rainfall, drought, wildfire, tropical cyclones), the impacts of recent extreme weather events of each type, and thus the degree to which various impacts are attributable to climate change. For instance, heat extremes have increased in likelihood and intensity worldwide due to climate change, with tens of thousands of deaths directly attributable. This is likely a significant underestimate due to the limited availability of impact information in lower- and middle-income countries. Meanwhile, tropical cyclone rainfall and storm surge height have increased for individual events and across all basins. In the North Atlantic basin, climate change amplified the rainfall of events that, combined, caused half a trillion USD in damages. At the same time, severe droughts in many parts of the world are not attributable to climate change. To advance our understanding of present-day extreme weather impacts due to climate change developments on several levels are required. These include improving the recording of extreme weather impacts around the world, improving the coverage of attribution studies across different events and regions, and using attribution studies to explore the contributions of both climate and non-climate drivers of impacts.