In large-scale epidemiological studies, it is increasing common to record physical activity objectively by wearable accelerometers. Accelerometry data are time series that allow more precise measurement of the intensity, frequency and duration of physical activity than self-reported questionnaires. However, standard analysis often reduce the high-resolution data into a few simple summary measures, which depends on choices of cut points and can be oversimplied. We develop a functional data framework for the analysis of accelerometry data. We first introduce functional indices to describe the profile of activity intensity, frequency and duration. These indices are then used as outcomes or predictors in functional regression analysis, which allows estimation of detailed dose-response relationship between activity patterns and health outcomes. These methods are motivated by and applied to the Objective Physical Activity and Cardiovascular Health Study among older women, where the aim is to study the association between objectively measured physical activity and cardiovascular diseases.