The devastating 2021 blackout in Texas, among others, has highlighted the need to reform electricity markets to make them resilient to extreme weather events. In this paper, we review related efforts by system planners and operators within electricity market contexts, focusing on Europe and the United States, and we analyze possible reforms to electricity capacity markets. To account for extreme weather events, capacity requirements and markets, along with other regulatory measures throughout the electricity and fuel supply chains, should be modified. First, capacity requirements must be tailored to the specific severe weather failure modes applicable to a given power system to achieve policymakers' reliability and resiliency objectives: reducing the frequency, magnitude and duration of blackouts. Second, all capacity requirements should be cost-effective and integrated with other non-capacity resources and requirements, such as transmission, distribution and other infrastructure systems. Third, for a capacity market to produce the desired efficiency benefits, the product (capacity) must be well-defined and backed by sufficient credit and other policies to ensure providers have sufficient incentives to perform when called.