Exploring complex energy landscape is a challenging issue in many applications. Besides locating equilibrium states, it is often also important to identify the transition states given by saddle points. In this talk, we will discuss the mathematics and algorithms, in particular, the shrinking dimer dynamics, developed to compute transition states. The method is simple to implement and does not require the explicit evaluation of Hessians. Some applications will be considered, including the study of critical nuclei morphology in solid state transformations, optimal photonic crystal design and the interacting particle systems.