Linear $\Omega$-algebras

In this paper we give a brief account of the basic results in the theory of linear $\Omega$-algebras. Particular attention is paid to research of recent years, and the connections of the theory of linear $\Omega$-algebras with other parts of algebra are shown. For some special cases of linear $\Omega$-algebras (ternary algebras, $\Gamma$-rings) only a survey of the literature is given.
With the help of linear $\Omega$-algebras new and simplified proofs of some known results in universal algebra are obtained. Various applications of linear $\Omega$-algebras to functional analysis and differential geometry are described.
A large number of open problems have been included, whose solution would apparently be of interest in the development of the theory of linear $\Omega$-algebras.