Protein repeats are considered hotspots of protein evolution, associated with acquisition of new functions and novel phenotypic traits, including disease. Paradoxically, however, repeats are often strongly conserved through long spans of evolution. To resolve this conundrum, it is necessary to directly compare paralogous (horizontal) evolution of repeats within proteins with their orthologous (vertical) evolution through speciation. Here we develop a rigorous methodology to identify highly periodic repeats with significant sequence similarity, for which evolutionary rates and selection (dN/dS) can be estimated, and systematically characterize their evolution. We show that horizontal evolution of repeats is markedly accelerated compared with their divergence from orthologues in closely related species. This observation is universal across the diversity of life forms and implies a biphasic evolutionary regime whereby new copies experience rapid functional divergence under combined effects of strongly relaxed purifying selection and positive selection, followed by fixation and conservation of each individual repeat.