The history of mathematics offers rich insights into both the evolution of human thoughts and the broader societal contexts. This talk will give a brief introduction to different research directions and challenges in the study of the history of mathematics. In particular, the speaker will delve into three main research approaches and their demands: internal history (the development of mathematics per se), external history (the social contexts of mathematics), and HPM (the application of history in mathematics education). The talk will also present examples of these approaches, main researchers in Taiwan, and an actual master’s thesis supervised by the speaker himself. Hopefully this talk will give graduate students a comprehensive overview about the history of mathematics as a research discipline.