In this study, we estimate the parameters of the Moyal distribution by using well-known and widely-used maximum likelihood (ML) and method of moments (MoM) methodologies. The ML estimators of the location and scale parameters of the Moyal distribution cannot be obtained in closed forms therefore iterative methods should be utilized. To make the study complete, modifed ML (MML) estimators for the location and the scale parameters of the Moyal distribution are also derived. The MML estimators are in closed forms and asymptotically equivalent to the ML estimators. Efficiencies of the MML estimators are compared with their ML and MoM counterparts using Monte-Carlo (MC) simulation study. Results of the simulation study show that the ML estimators are more efficient than the MML and MoM estimators for small sample sizes. However when the sample size increases performances of the ML and MML estimators are almost same in terms of the Defficiency (Def) criterion as expected. At the end of the study, a real data set is used to show the implementation of the methodology developed in this paper.