We study a hard geometric problem. Given n points in the plane and a positive integer k, the Rectilinear k -Bends Traveling Salesman Problem asks if there is a piecewise linear tour through the n points with at most k bends where every line-segment in the path is either horizontal or vertical. The problem has applications in VLSI design. We prove that this problem belongs to the class FPT (fixed-parameter tractable). We give an algorithm that runs in O(kn 2?+?k 4k n) time by kernelization. We present two variations on the main result. These variations are derived from the distinction between line-segments and lines. Note that a rectilinear tour with k bends is a cover with k line-segments, and therefore a cover by lines. We derive FPT-algorithms using bounded-search-tree techniques and improve the time complexity for these variants.