We present a review of our works on development of efficient numerical algorithms of finding transition states on the potential energy surfacne and free energy surface. The transition state arises from the noise-induced rare transition events in a gradient system associated with a smooth high dimensional function with multiple wells. As the lowest-energy unstable point connecting two neighboring local minimizers(wells), the transition state has the Morse index one -- defined as only one unstable eigenvector. We review the Gentlest Ascent Dynamics (Nonlinearity 2011)and Interative Minimization Algorithm (SINUM 2015,JCP2015) as well as the extended development for this long-term challenging problem in computational chemistry and biology. The key component is to recursively identify the lowest eigenmode of the Hessians for in Euclidan or Hilbert space. Some recent work based on the new approach of Witten Laplacian may be also introduced if time permits. These works are colloboratively joint with Weiguo Gao, Shuting Gu, Hongqiao Wang, Xiaoguang Li.