Functions ^NF (x₁,...,x_N) which are a straightforward generalisation of standard hypergeometric functions of N variables are introduced. A convenient operator representation is established for these functions which permits one to consider, in a uniform and simple way, standard hypergeometric functions (generalised hypergeometric series of one variable, Appell, Kampe de Feriet and Lauricella functions, etc.) as well as many other nameless hypergeometric series arising in physical and quantum chemical applications. Generating functions, generalisations of Bailey (1935) and Chaundy (1943) expansions, recurrence relations, reduction rules and some other topics are discussed.