An absolutely representing system (ARS) in a Banach spaceXis a setD⊂Xsuch that every vectorxinXadmits a representation by an absolutely convergent seriesx= Σiaixiwith (ai) ⊂Rand (xi) ⊂D. We investigate some general properties of absolutely representing systems. In particular, absolutely representing systems in uniformly smooth and in B-convex Banach spaces are characterized viaε-nets of the unit balls. Every absolutely representing system in a B-convex Banach space is quick, i.e., in the representation above one can achieve ∥aixi∥ <cqi∥x∥,i= 1, 2,… for some constantsc> 0 andq∈ (0,1).