Climate change and natural disasters caused by hydrological, meteorological, and climatic causes have a significant and increasing direct and indirect impact on human health, leading to increased mortality and morbidity. Russia is a country that suffers from frequent climatic and weather disasters. This is mainly due to its vast territory, complex geographical and ecological environment, and widely varying climatic conditions. This review provides information on climatological and hydrological extremes in Russia in 2010–2020, floods and droughts, and their impact on the health and well-being of the country’s population. A literature search was conducted using electronic databases Web of Science, Pubmed, Science Direct, Scopus, and e-Library, focusing on peer-reviewed journal articles published in English and in Russian from 2010 to 2021. Four conceptual categories were used: “floods”, “droughts”, “human health”, and “Russia”. It is concluded that while most hazardous weather events cannot be completely avoided, many health impacts can potentially be prevented. The recommended measures include early warning systems and public health preparedness and response measures, building climate resilient health systems and other management structures.