Let (Y_n,V_n) be i.i.d. distributed, with the components r and s-dimensional, respectively. Reflected random walk starting at a point x of the positive r-dimensional orthant is defined recursively by X_0 = x, X_n = |X_{n−1}−Y_n|, where |(a_1,...,a_r)| = (|a_1|,...,|a_r|). In R^s, consider the ordinary sum S_n = V_1 +···+V_n . We are interested in (topological) recurrence of the process (X_n,v+S_n) starting at (x,v). While this is quite well understood for refelcted random walk with r=1, in higher dimension (r \geq 2) or with some non-reflected coordinates (s \in {1,2}) we have a few basic results and various open questions with some partial answers. This is work with Judith Kloas, with input from Marc Peigne' and Wojciech Cygan.