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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 (λ; κ) ν and [λ; κ] ν, we introduce here the family of the incomplete generalized τ-hypergeometric functions 2γτ 1 (z) and 2Γτ 1 (z). 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 τ-Appell hypergeometric functions Γτ1, τ2