Random walks and diffusing particles have been a corner stone in modelling the random motion of a varying quantity with applications spanning over many research fields. And in most of the applications one can ask a question related to when something happened for the first time. That is, a first-passage problem. Typical examples include chemical reactions which can not happen until the constituents meet for the first time, a neuron firing when a fluctuating voltage exceeds a threshold value and the triggering of buy/sell orders of a stock option. The applications are many, which is why first-passage problems have attracted researchers for a long time, and will keep doing so. In this thesis we analyse first-passage problems analytically.

A stochastic system can always be simulated, so why bother with analytical solutions? Well, there are many system where the first passage is improbable in a reasonable time. Simulating those systems with high precision is hard to do efficiently. But evaluating an analytical expression happens in a heart beat. The only problem is that the first-passage problem is tricky to solve as soon as you take a small step away from the trivial ones. Consequently, many first-passage problems are still unsolved.

In this thesis, we derive approximate solutions to first-passage related problems for a random walker and a diffusing particle bounded in a potential, which the current methods are unable to handle. We also study a continuous-time random walker on a network and solve the corresponding first-passage problem exactly in way that has not been done before. These results give access to a new set of analytical tools that can be used to solve a broad class of first-passage problems.