Hawkes processes have been widely used in many areas, but their probability properties such as closed-forms for moments can be quite difficult. In the present talk, some simple introduction to Hawkes processes is given first, then the approaches to obtain moments of linear Hawkes processes and/or the intensity of a number of marked linear Hawkes processes are presented. In particular, the elementary method given by Cui et al. (2020) is discussed in details. This method works not only for all Markovian Hawkes processes, but also for some non-Markovian Hawkes processes. The method is applied for one-dimensional linear Hawkes processes and other related processes such as Cox processes, dynamic contagion processes, non-homogenous Poisson processes and non-Markovian cases (Gamma decay intensity). Several results are obtained which may be useful in studying Hawkes processes and other counting processes. This proposed method is an extension of Dynkin’s formula, which is simple and easy to use. Finally, our most recent work (Cui & Sornette) on moments for linear and nonlinear Hawkes processes is simply introduced by using stochastic partial differential equation method given by Kanazawa & Sornette (2020, 2021).