In the present paper, we consider Korovkin and quantitative theorems, which have been treated by various authors to date, under weighted approximation. After giving the basic definitions and some of well-known spaces, we mention the main theorems and their applications to linear positive operators, which have been specially treated by the authors. Therefore, this study which can be considered as a survey study will direct the readers to literature information. Furthermore, we give a general operator including well-known operators as an application of some theorems given at the end of the paper.