High-dimensional random functionals emerge ubiquitously when modeling complex systems: as energy landscapes in physics, fitness landscapes in biology, loss landscapes in machine learning, to mention a few examples. They are typically very non-convex, with a high number of local minima, local maxima and saddles with different stability properties. In this talk, I will discuss how to use tools from random matrix theory to gain information on the statistical distribution of the saddles in a prototypical random landscape with Gaussian statistics, and I will briefly comment on how to use this information to characterize how the functional is explored dynamically.