We introduce the Romu family of pseudo-random number generators (PRNGs) which combines the nonlinear operation of rotation with the linear operations of multiplication and (optionally) addition. Compared to conventional linear-only PRNGs, this mixture of linear and nonlinear operations achieves a greater degree of randomness using the same number of arithmetic operations. Or equivalently, it achieves the same randomness with fewer operations, resulting in higher speed. The statistical properties of these generators are strong, as they pass BigCrush and PractRand -- the most stringent test suites available. In addition, Romu generators take maximum advantage of instruction-level parallelism in modern superscalar processors, giving them an output latency of zero clock-cycles when inlined, thus adding no delay to an application. Scaled-down versions of these generators can be created and tested, enabling one to estimate the maximum number of values the full-size generators can supply before their randomness declines, ensuring the success of large jobs. Such capacity-estimates are rare for conventional PRNGs. A linear PRNG has a single cycle of states of known length comprising almost all possible states. However, a Romu generator computes pseudo-random permutations of those states, creating multiple cycles with pseudo-random lengths which cannot be determined by theory. But the ease of creating state-sizes of 128 or more bits allows (1) short cycles to be constrained to vanishingly low probabilities, and (2) thousands of parallel streams to be created having infinitesimal probabilities of overlap.