Abstarct: In this thesis, we first study variants of the classical geometric shortest path problem inside a simple polygon. In this version we allow to go outside the polygon .Let P be a simple polygon with n vertices and let s and t be two points in P. For an integer k ≥ 0, we define a k-violation path from s to t to be a connected path between s and t whose intersection with outside P consists of at most k segments. The number of segments of the path inside P. has no restriction The goal is to compute a path with minimum Euclidean length among all (≤k)-violation paths from s to t. In this thesis, we study this problem for k = 1 and give an algorithm to compute the shortest one-violation path in O(n^3) time. We show that this problem for rectilinear polygons, is solvable in O(n log n) time. Then, we study the problem of computing shortest paths in the plane among h convex obstacles, where the path is allowed to go through (violate) up to k obstacles, for k ≤ h. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of θ(kn) on the size of this map, and show that it can be computed in O(k^2 n log⁡n) time, where n is the total number of obstacle vertices. Finally, we will study visibility graphs, which will be used to compute the length of the shortest path between two points inside a given polygon..