Sensing and imaging systems are under increasing pressure to accommodate ever-larger and higher-dimensional data sets; ever-faster capture, sampling, and processing rates; ever-lower power consumption; ever-smaller form factor; and new sensing modalities. These needs have motivated the development of new approaches to signal acquisition and processing. We provide an introduction to the field of compressive sensing (CS), which has stimulated a rethinking of sensor and signal processing system design. In CS, analog signals are digitized and processed not via uniform sampling but via measurements using more general, even random, test functions. In contrast to conventional wisdom, the new theory asserts that one can combine “sub-Nyquist rate sampling” with large-scale optimization for efficient and accurate signal acquisition when the signal has a sparse structure. Particular topics addressed include signal sparsity, randomized sampling, optimization-based signal recovery, and perspectives on applications to seismic data acquisition and processing.