Many seismic exploration techniques rely on the collection of massive data volumes that are subsequently mined for information during processing. Although this approach has been extremely successful in the past, current efforts toward higher-resolution images in increasingly complicated regions of the earth continue to reveal fundamental shortcomings in our workflows. Chiefly among these is the so-called “curse of dimensionality” exemplified by Nyquist's sampling criterion, which disproportionately strains current acquisition and processing systems as the size and desired resolution of our survey areas continue to increase. We offer an alternative sampling method leveraging recent insights from compressive sensing toward seismic acquisition and processing for data that are traditionally considered to be undersampled. The main outcome of this approach is a new technology where acquisition and processing related costs are no longer determined by overly stringent sampling criteria, such asNyquist. At the heart of our approach lies randomized incoherent sampling that breaks subsampling related interferences by turning them into harmless noise, which we subsequently remove by promoting transform-domain sparsity. Now, costs no longer grow significantly with resolution and dimensionality of the survey area, but instead depend only on transform-domain sparsity. Our contribution is twofold. First, we demonstrate by means of carefully designed numerical experiments that compressive sensing can successfully be adapted to seismic exploration. Second, we show that accurate recovery can be accomplished for compressively sampled data volumes sizes that exceed the size of conventional transform-domain data volumes by only a small factor. Because compressive sensing combines transformation and encoding by a single linear encoding step, this technology is directly applicable to acquisition and to dimensionality reduction during processing. In either case, sampling, storage, and processing costs scale with transform-domain sparsity. We illustrate this principle by means of number of case studies.