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This reprint contains 10 papers published in a Special Issue of the MDPI journal Mathematics concerned with hypercompositional algebra. The contributions, written by 24 authors from 10 different countries, cover a wide range of topics concerning the theoretical aspects, applications, and computational issues of algebraic hyperstructures. The reader will find a wide range of problems, arguments and methods that illustrate and provide insight into the variegated features of this theory. This reprint will be of interest to those working in abstract algebra, hypergroup theory, finite geometries or combinatorics, as well as to those who wish to familiarise themselves with recent advances in hypercompositional algebra.