  We define cut-and-join operator in Hurwitz theory for merging of two
branching points of arbitrary type. These operators have two alternative
descriptions:(i) they have the GL characters as eigenfunctions and the
symmetric-group characters as eigenvalues; (ii) they can be represented as
differential operators of the $W$-type (in particular, acting on the
time-variables in the Hurwitz-Kontsevich tau-function). The operators have the
simplest form if expressed in terms of the matrix Miwa-variables. They form an
important commutative associative algebra, a Universal Hurwitz Algebra,
generalizing all group algebra centers of particular symmetric groups which are
used in description of the Universal Hurwitz numbers of particular orders. This
algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear
form on the linear space of Young diagrams, evaluated at the product of all
diagrams, which characterize particular ramification points of the covering.
