The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of compound Poisson, Hawkes, shot-noise Poisson and dynamic contagion process. As an application of the dynamic contagion process, we illustrate that this point process can be used to count cumulated number of infections (e.g. the COVID-19 infections) arising in a country from contagious catastrophic events with numerical examples.