Reflection seismic data continuation is the computation of data at source and receiver locations that differ from those in the original data, using whatever data are available. We develop a general theory of data continuation in the presence of caustics and illustrate it with three examples: dip moveout (DMO), azimuth moveout (AMO), and offset continuation. This theory does not require knowledge of the reflector positions. We construct the output data set from the input through the composition of three operators: an imaging operator, a modeling operator, and a restriction operator. This results in a single operator that maps directly from the input data to the desired output data. We use the calculus of Fourier integral operators to develop this theory in the presence of caustics. For both DMO and AMO, we compute impulse responses in a constant-velocity model and in a more complicated model in which caustics arise. This analysis reveals errors that can be introduced by assuming, for example, a model with a constant vertical velocity gradient when the true model is laterally heterogeneous. Data continuation uses as input a subset (common offset, common angle) of the available data, which may introduce artifacts in the continued data. One could suppress these artifacts by stacking over a neighborhood of input data (using a small range of offsets or angles, for example). We test data continuation on synthetic data from a model known to generate imaging artifacts. We show that stacking over input scattering angles suppresses artifacts in the continued data.