In this talk we briefly discuss different types of symmetries in probability, including stationarity, stationarity of the increments, isotropy, self-similarity and their combinations. Each of these symmetries can be expressed as the invariance of the distribution of the stochastic processes with respect to certain action. In particular, we consider the random locations of stochastic processes, such as the hitting times or the location of the path supremum over a fixed interval. On one hand, we see how the probabilistic symmetries can imply the properties of the distributions of these random locations; on the other hand, we also discuss how the distributions of the random locations can be used to characterize the probabilistic symmetries. We then proceed to propose a unified framework to study the random locations for stochastic processes exhibiting probabilistic symmetries.