Abstract Efficiently answering reachability queries on a directed graph is a fundamental problem and many solutions – theoretical and practical – have been proposed. A common strategy to make reachability query processing efficient, accurate and scalable is to precompute indexes on the graph. However this often becomes impractical, particularly when dealing with large graphs that are highly dynamic or when queries have additional constraints known only at the time of querying. In the former case, indexes become stale very quickly and keeping them up- to-date at the same speed as changes to the graph is untenable. For the latter setting, currently proposed indexes are often quite bulky and are highly customized to handle only a small class of constraints. In this paper, we propose a first practical attempt to address these issues by abandoning the traditional indexing approach altogether and operating directly on the graph as it evolves. Our approach, called ARROW, uses random walks to efficiently approximate reachability between vertices, building on ideas that have been prevalent in the theory community but ignored by practitioners. Not only is ARROW well suited for highly dynamic settings – as it is index-free, but it can also be easily adapted to handle many different forms of ad-hoc constraints while being competitive with custom-made index structures.