Many classic tasks in vision -- such as the estimation of optical flow or stereo disparities -- can be cast as dense correspondence matching. Well-known techniques for doing so make use of a cost volume, typically a 4D tensor of match costs between all pixels in a 2D image and their potential matches in a 2D search window. State-of-the-art (SOTA) deep networks for flow/stereo make use of such volumetric representations as internal layers. However, such layers require significant amounts of memory and compute, making them cumbersome to use in practice. As a result, SOTA networks also employ various heuristics designed to limit volumetric processing, leading to limited accuracy and overfitting. Instead, we introduce several simple modifications that dramatically simplify the use of volumetric layers - (1) volumetric encoder-decoder architectures that efficiently capture large receptive fields, (2) multi-channel cost volumes that capture multi-dimensional notions of pixel similarities, and finally, (3) separable volumetric filtering that significantly reduces computation and parameters while preserving accuracy. Our innovations dramatically improve accuracy over SOTA on standard benchmarks while being significantly easier to work with - training converges in 7X fewer iterations, and most importantly, our networks generalize across correspondence tasks. On-the-fly adaptation of search windows allows us to repurpose optical flow networks for stereo (and vice versa), and can also be used to implement adaptive networks that increase search window sizes on-demand.