Besides the classical algebraic structures of groups, rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling, other algebraic structures such as dioïds and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non-classical problems, e.g. in Decision Analysis, Fuzzy Set Theory, Operations Research, Automatic Control and Mathematical Physics. The present paper aims at providing an overview of applications of dioïd and semiring structures, stressing links with Fuzzy Sets and emphasizing linear algebraic problems (solving linear systems, computing eigenvalues and eigenvectors), non-classical path-finding problems (using algebras of endomorphisms) and connections between dioïd structure and nonlinear analysis (in view of solving problems in Mathematical Physics).