Gjergji, there are remarkable articles by Dick Richard Askey, on :
(1) "Ramanujan and hypergeometric functionsand basic hypergeometric series"
in
Russian
Math. Surveys 45:1 (1990) 37--86;
reprinted in Ramanujan: essays and another one I have to find after surveys, Hist. Math. 22
Amer. Math. Soc., Providence, RI, 2001, pp. 277--324;
(2) "A look at the dinnerBateman project" in The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), 29--43, Contemp. Math. 169, Amer. Math. Soc., Providence, RI, 1994.
(I asked Dick exactly this question, maybe without accenting on "modern theory", some years ago.) The modern theory is mostly multiple hypergeometric functions related to root systems; for a nice survey on the roots of these functions, the Selberg integral, see
(3) P. Forrester and S.O. Warnaar, "The importance of the Selberg integral", Bull. Amer. Math. Soc. (N.S.) 45:4 (2008) 489--534.
Elliptic functions are hypergeometric functions of the 21st century:
(4) V.P. Spiridonov, "Essays on the theory of elliptic hypergeometric functions", Russian Math. Surveys 63:3 (2008) 405--472.
