Entanglement is thought to be the critical resource in quantum computing and quantum communication. We explain how this physical concept is related to ideas and problems in mathematics, and particularly in functional analysis, convex analysis and high- dimensional geometry. This allows to show that the phenomenon of entanglement is ubiquitous: for example, for 8 qubits (one may say, a qubyte), the proportion of the states that are entanglement-free is smaller than 10^(-19990), when measured by the standard Euclidean volume. One may similarly compare various classes of quantum maps (or channels), related with dynamical changes of the physical system.