The basic concept of the multiobjective optimization evolutionary algorithm based on decomposition (MOEA/D) is to transform a multiobjective optimization problem into a set of subproblems (single-objective or multiobjective) for optimization solutions. Since MOEA/D was proposed in 2007, it has attracted extensive attention from Chinese and international scholars and become one of the most representative multiobjective optimization evolutionary algorithms. This study summarizes the research progress on MOEA/D in the past thirteen years. The advances include algorithm improvements of MOEA/D, research of MOEA/D on many-objective optimization and constraint optimization,and application of MOEA/D in some practical issues. Then, several representative improved algorithms of MOEA/D are compared through experiments. Finally, the study presents several potential research topics of MOEA/D in the future.