In this progress report, recent advances in the development of organic transistors with superior bias stress stability and in the understanding of the charge traps that degrade device performance under prolonged bias stress are reviewed, with a particular focus on materials science and engineering methods. The phenomenological aspects of bias stress effects and the experimental methods for investigating charge traps are described. The recent progress in the bias stress stability of organic transistors is discussed in terms of those components that are the main focus of attempts to improve bias stress stability, i.e., organic semiconductor layers, gate dielectrics, and source/drain contacts. A brief summary of this progress is presented and the outlook for future research in this field is assessed. This report aims to summarize recent progress in this field and to provide some guidelines for studying bias stress-induced charge-trapping phenomena.