We present several new effcient algorithms for approximating the girth, g, of weighted and unweighted n-vertex, m-edge undirected graphs. For undirected graphs with polynomially bounded, integer, non-negative edge weights, we provide an algorithm that for every integer k 1, runs in eO(m + n1+1=k log g) time and returns a cycle of length at most 2kg. For unweighted, undirected graphs we present an algorithm that for every k 1, runs in eO (n1+1=k) time and returns a cycle of length at most 2kdg=2e, an almost k-approximation. Both algorithms provide trade-off-s between the running time and the quality of the approximation. We also obtain faster algorithms for approximation factors better than 2, and improved approximations when the girth is odd or small (e.g., 3 and 4).