The basic idea 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). Since MOEA/D was proposed in 2007, it has attracted extensive attention from scholars all over the world. MOEA/D has become one of the most representative multiobjective optimization evolutionary algorithms. This paper summarizes the research progress on MOEA/D in the past thirteen years. It includes: (1) the improvements of MOEA/D; (2) the research of MOEA/D on many-objective optimization problems and constrainted optimization problems; (3) the application of MOEA/D on some real-world problems. Then, this paper compares experimentally the performance of several representative improved algorithms of MOEA/D. Finally, this paper presents several potential research topics of MOEA/D in the future.