The multi-objective simulation optimization (MOSO) problem is a nonlinear multi-objective optimization problem in which multiple simultaneous and conflicting objective functions can only be observed with stochastic error. We provide an introduction to MOSO at the advanced tutorial level, aimed at researchers and practitioners who wish to begin working in this emerging area. Our focus is exclusively on MOSO methods that characterize the entire efficient or Pareto optimal set as the solution to the MOSO problem; this set then may be used as input to the broader multi-criteria decision-making process. Our introduction to MOSO includes an overview of existing theory, methods, and provably convergent algorithms for (a) MOSO on finite sets, called multi-objective ranking and selection, (b) MOSO with integer-ordered decision variables, and (c) MOSO with continuous decision variables. In the context of integer-ordered and continuous decision variables, we focus on methods that provably converge to a local efficient or Pareto optimal set under the natural ordering. We also discuss key open questions that remain in this emerging field.