Schur functions have been used to compute characters of the symmetric group and general linear Lie algebra. Weyl generalized Schur's theory to other classical groups via a generalized Jacobi-Trudi identity which
has turned into one chapter in his seminal book on classical groups. Recently, Okounkov and Olshanski used a difference analog of Schur functions to compute the Harish-Chandra central character for the general linear Lie group. We will discuss a generalized Schur functions that take all these important variant Schur functions under one umbralla and derive several identities for the new functions. Joint work with N. Rozhkovskaya.