By e(p)(r), we mean the space of all sequences whose Euler transforms of order r are in the sequence spaces l(p) and l(infinity) (see [B. Allay, F. Basar, M. Mursaleen, On the Euler sequence spaces which include the spaces l(p) and l infinity I, Inform. Sci. (2005) (in press)]), where 1 < p < infinity. In the present paper, we essentially characterize the classes (e(p)(r) : l infinity), (e(l)(r) : l(p)) and (e(p)(r) : f) of infinite matrices for 1 < p <= infinity and give the characterizations of some other matrix mappings from the space er to the Euler, Riesz, difference, etc., sequence spaces, by means of a given basic lemma. We devote the final section of the paper to examining some geometric properties of the space e(p)(r). (c) 2005 Elsevier Ltd. All rights reserved.