Understanding the influence of solar variability on the Earth's climate requires knowledge of solar variability, solar-terrestrial interactions, and the mechanisms determining the response of the Earth's climate system. We provide a summary of our current understanding in each of these three areas. Observations and mechanisms for the Sun's variability are described, including solar irradiance variations on both decadal and centennial time scales and their relation to galactic cosmic rays. Corresponding observations of variations of the Earth's climate on associated time scales are described, including variations in ozone, temperatures, winds, clouds, precipitation, and regional modes of variability such as the monsoons and the North Atlanti...