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Rayner rngs are rngs (rings without unity) whose elements are formal power series whose coefficients lie in a rng and the exponents lie in an additive ordered group, such that the supports of these series belong to a predetermined ideal constrained by a set of axioms. The work presents an inspection of the interplay between the algebraic, topological and categorical properties of the Rayner rngs, the rngs of coefficients and the ordered groups of exponents, studying the Rayner rngs under varied theoretical perspectives and seeking universal relations between them. Two key topologies on these structures are systematically analysed, the so-called weak and strong topologies, and a version of the Intermediate Value Theorem is obtained for the weak topology. Special attention is given to rngs of Levi-Civita, Puiseux and Hahn series, which are prominent instances of Rayner rngs.