Given a set of disjoint polygonal obstacles in the plane, the problem of processing approximate shortest path queries is that of reporting an obstacle-avoiding path $P$ (or its length) between two arbitrary query points $p$ and $q$ in the plane, such that the length of $P$ is within a small constant factor of the length of the shortest obstacle-avoiding path between $p$ and $q$. The goal is to answer each approximate shortest path query quickly by constructing data structures that capture path information about the obstacle-scattered environment. In this talk, we present a family of efficient algorithms for this problem that demonstrate an interesting trade-off between the approximation factor, query time, and time and space bounds of the data structures. Our main results are algorithms that achieve logarithmic length query time, after building data structures in subquadratic time and space. Our algorithms are based on several new ideas, such as constructing planar spanners with Steiner points and answering approximate shortest path queries on planar graphs.