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A fundamental aspect of a random walk is: when does it reach a specified position for the first time? This is the first-passage time. More generally, the distribution of first passage times underlies many non-equilibrium phenomena, such as the triggering of integrate-and-fire neurons, the statistics of cell division, and the execution of stock options. The computation of the first-passage time and its distribution is both simple and beautiful, with profound connections to electrostatic potential theory. I will present some aspects of these fundamental ideas. I'll also present the backward Kolmogorov approach that allows one to compute the first-passage time by solving a time-independent equation. This approach will be illustrated for a diffusing particle in a finite interval.