Vector optimization is one of the important research aspects in the field of mathematical programming and some related fundamental theory and basic methods have great theoretical significance and applicable value. In recent years, study on the definitions and properties of approximate solutions has become one research hotspot in theory and methods of vector optimization. In this paper, we mainly review some important progress about some concepts of approximate solutions and unified solutions, and their development and some properties of various kinds of approximate solutions and unified solutions for a class of vector optimization problems by Chinese researchers and especially for our group. In the end, we also propose some open questions related to approximate solutions and unified solutions in vector optimization problems.