Analytic function methods in the complex plane for plane potential fields inspire searching analogous effective methods for solving spatial and multidimensional problems of mathematical physics. Many such methods are based on mappings of hypercomplex algebras. An idea of an algebraic-analytic approach to elliptic equations of mathematical physics means a finding of commutative Banach algebra such that differentiable functions with values in this algebra have components satisfying the given equation with partial derivatives. An use of differentiable functions given in commutative Banach algebras combines the preservation of basic properties of analytic functions of a complex variable for the mentioned differentiable functions and the convenience and the simplicity of construction of solutions of PDEs. The paper contains the review of results reflecting the formation and the development of the mentioned approach.

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