Eigenvalues' coalescence has attracted great attention for quite some time. Besides having important implications in Physics, Chemistry and Engineering, it is also a subject of remarkable mathematical interest and beauty. In this talk we will overview the main results that concern eigenvalues' coalescence for real symmetric and complex Hermitian matrix functions, and address the problem of how one can numerically detect and approximate the points in parameters' space where the coalescence occur. The talk will be based on joint work with Luca Dieci (Georgia Institute of Technology, USA), Alessandra Papini (University of Florence, Italy) and Alessandro Spadoni (Dept. of Mechanical Engineering, EPFL, Switzerland).