Saddle-points and mountain-pass points of energy surfaces have many applications in areas that range from plate buckling and interface propagation, to theoretical chemistry and image processing. This thesis starts by giving a motivation for the subject. In Chapter 2 we then present an overview of the most common algorithms for locating saddle-points. Chapter 3 presents a novel class of algorithms that rely on applying parameter continuation to a constraint minimization problem. We test the method on various two-dimensional surfaces, and convergence rates are computed for a basic quadratic example. In Chapter 4 we discuss the convergence of the string method in infinitc-dimcnsions, which wc then apply in Chapters 5 and 6 to two different PDEs, respectively a p-Laplacian operator with a double-well potential and a fourth order PDE linked to a buckling problem. We observe interesting behaviors of the String Method such as location of multiple critical points and saddle-continua. New kinds of solutions are found in both PDEs, which makes analysis of the equations themselves interesting in their own right. Finally, a survey of different algorithms is presented in Chapter 7, based both on the literature and numerical experiments from this thesis, followed by a conclusion and directions for future work.