Abstract: This talk will introduce the concept of G complete-reducibility (G c-r) originally conceived by Serre in the 90s. This idea has important connections to the open problem of classifying the reductive subgroups. Significant progress has been made in this classification by many authors over fields of larger characteristics. I will give some background on linear algebraic groups and explain the methodology of the classification so far and what still needs to be done. I will explain the importance of studying reductive subgroups and some results I have found in small characteristics.