Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols

and

, we introduce here the family of the incomplete generalized

-hypergeometric functions

and

. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second

-Appell hypergeometric functions

and

of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.