  Association schemes form one of the main objects of algebraic combinatorics,
classically defined on finite sets. In this paper we define association schemes
on arbitrary, possibly uncountable sets with a measure. We study operator
realizations of the adjacency algebras of schemes and derive simple properties
of these algebras. To develop a theory of general association schemes, we focus
on schemes on topological Abelian groups where we can employ duality theory and
the machinery of harmonic analysis. We construct translation association
schemes on such groups using the language of spectrally dual partitions. Such
partitions are shown to arise naturally on topological zero-dimensional Abelian
groups, for instance, Cantor-type groups or the groups of p-adic numbers. This
enables us to construct large classes of dual pairs of association schemes on
zero-dimensional groups with respect to their Haar measure, and to compute
their eigenvalues and intersection numbers. We also derive properties of
infinite metric schemes, connecting them with the properties of the
non-Archimedean metric on the group.
  Pursuing the connection between schemes on zero-dimensional groups and
harmonic analysis, we show that the eigenvalues have a natural interpretation
in terms of Littlewood-Paley wavelet bases, and in the (equivalent) language of
martingale theory. For a class of nonmetric schemes constructed in the paper,
the eigenvalues coincide with values of orthogonal functions on
zero-dimensional groups. We observe that these functions, which we call
Haar-like bases, have the properties of wavelets on the group, including in
some special cases the self-similarity property. This establishes a seemingly
new link between algebraic combinatorics and harmonic analysis.
  We conclude the paper by studying some analogs of problems of classical
coding theory related to the theory of association schemes.
