This tutorial and review of multi-objective optimization (MOO) gives a detailed explanation of the 5 steps to create, solve, and then select the optimum result. Unlike single-objective optimization, the fifth step of selection or ranking of solutions is often overlooked by the authors of papers dealing with MOO applications. It is necessary to undertake a multi-criteria analysis to choose the best solution. A review of the recent publications using MOO for chemical process engineering problems shows a doubling of publications between 2016 and 2019. MOO applications in the energy area have seen a steady increase of over 20% annually over the last 10 years. The three key methods for solving MOO problems are presented in detail, and an emerging area of surrogate-assisted MOO is also described. The objectives used in MOO trade off conflicting requirements of a chemical engineering problem; these include fundamental criteria such as reaction yield or selectivity; economics; energy requirements; environmental performance; and process control. Typical objective functions in these categories are described, selection/ranking techniques are outlined, and available software for MOO are listed. It is concluded that MOO is gaining popularity as an important tool and is having an increasing use and impact in chemical process engineering.