The determination of maximum likelihood estimators for the three–parameter Weibull model is usually considered a nontrivial problem, because of the nonlinear likelihood equations. Despite the availability of a large number of algorithms that tackle this problem, there is considerable dissatisfaction among practitioners, who report an inability to conveniently determine the desired MLE's (e.g. Adatia and Chan, 1985; Sinha and Sloan, 1988). We therefore review several existing algorithms for obtaining these estimators and investigate the causes of failure and some computational difficulties associated with these procedures. We find that a simple procedure outlined by Lawless (1982) is the only one that is guaranteed to yield parameter estimates that maximize the likelihood function for any sample. We describe this procedure in detail with modifications to improve its efficiency. The method is based on first principles and requires less computational effort than most other schemes. Several examples from the literature are used to illustrate the difficulties encountered by other methods.