Summary: The difference sequence spaces

${\ell}_{\infty}\left({\Delta}\right)$,

$c\left({\Delta}\right)$ and

${c}_{0}\left({\Delta}\right)$ were studied by

*H. Kizmaz* [Can. Math. Bull. 24, 169–176 (1981;

Zbl 0454.46010)]. The sequence spaces

${a}_{0}^{r}$ and

${a}_{c}^{r}$ have been recently defined and examined by

*C. Aydin* and

*F. Basar* [Hokkaido Math. J. 33, No. 2, 383–398 (2004;

Zbl 1085.46002)]. The main purpose of the present paper is to introduce the spaces

${a}_{0}^{r}\left({\Delta}\right)$ and

${a}_{c}^{r}\left({\Delta}\right)$ of difference sequences. Moreover, it is proven that the spaces

${a}_{0}^{r}\left({\Delta}\right)$ and

${a}_{c}^{r}\left({\Delta}\right)$ are BK-spaces including the spaces

${c}_{0}$ and

$c$, and some inclusion relations are given. It is also proven that the sequence space

${a}_{0}^{r}$ has the AD property while the space

${a}_{0}^{r}\left({\Delta}\right)$ does not. Furthermore, a basis and the

$\alpha $-,

$\beta $- and

$\gamma $-duals of the spaces

${a}_{0}^{r}\left({\Delta}\right)$ and

${a}_{c}^{r}\left({\Delta}\right)$ are determined. The last section of the paper is devoted to characterizations of the matrix classes

$({a}_{c}^{r}\left({\Delta}\right):{\ell}_{p})$ and

$({a}_{c}^{r}\left({\Delta}\right):c)$, and the characterizations of some other matrix classes are obtained by means of a given basic lemma, where

$1\le p\le \infty $.