The Hawkes process is a self-exciting Poisson point process, characterised by a conditional intensity function. Since its introduction fifty years ago, it has been the subject of numerous research directions and continues to inspire new methodological and theoretical developments as well as new applications. This paper marks half a century of interest in Hawkes processes by presenting a snapshot of four state-of-the-art research directions, categorised as frequentist and Bayesian methods, other modelling approaches and notable theoretical developments. A particular focus is on nonparametric approaches, with advances in kernel estimation and computational efficiencies. A survey of real world applications is provided to illustrate the breadth of application of this remarkable approach.