A sound field on a line or in a plane has an effectively limited spatial bandwidth determined by the temporal frequency. Similar can be said for sound fields from far-field sources when analyzed on circular and spherical apertures. Namely, for a given frequency and aperture size, a sound field is effectively composed of a finite number of circular or spherical harmonic components. Based on these two observations, it follows that if adequately sampled, sound fields can be represented and manipulated in a digital domain with negligible loss of information. The optimal sampling surface depends on the problem geometry, and the set of sampling points needs to be in accordance with the Nyquist criterion relative to the mentioned effective sound field bandwidth. In this thesis, we address the problems of sound field capture and reproduction from a practical perspective. More specifically, we present approaches that do not depend on acoustical models, but rely instead on obtaining an acoustic MIMO channel between transducers (microphones or loudspeakers) and a set of sampling (or control) points. Subsequently, sound field capture and reproduction are formulated as constrained optimization problems in a spatially discrete domain and solved using conventional numerical optimization tools. The first part of the thesis deals with spatial sound capture. We present a framework for analyzing and designing differential microphone arrays based on spatiotemporal sound field gradients. We also show how to record two- and three-dimensional sound fields with differential, circular, and spherical microphone arrays. Finally, we use the mentioned discrete optimization for computing filters for directional and sound field microphone arrays. In the second part of the thesis, we focus on spatial sound reproduction. We first present a design of a baffled loudspeaker array for reproducing sound with high directivity over a wide frequency range, which combines beamforming at low, and scattering from a rigid baffle at high frequencies. We next present Sound Field Reconstruction (SFR), which is an approach for optimally reproducing a desired sound field in a wide listening area by inverting a discrete, MIMO acoustic channel. In the end, we propose a single- and multi-channel low-frequency room equalization method, formulated as a discrete constrained optimization problem, with constraints designed to prevent excessive equalization filter gains, localization bias, and temporal distortions in the form of pre- and post-echos.