We propose a family of randomized algorithms for undirected s-t-connectivity which achieve a time-space product of $S\cdot T = \tilde O(n^2)$ for a graph with $n$ nodes and $m$ edges (where the $\tilde O$-notation disregards poly-logarithmic terms). In particular, we obtain a log-space algorithm which solves s-t-connectivity faster than the random walk, as well as an algorithm running in time $\O(n+m)$ which is, in general, more space-efficient than BFS or DFS. The algorithms rely on a new Monte-Carlo type walk on graphs, which is then combined with the landmark-based scheme of Broder et al. (1994).