In this seminar, we will examine the landmark philosophical works on the concept of scientific explanation. We will begin with Hempel and Oppenheim's (1945) work on two covering law accounts of scientific explanation. For better or for worse, most of the subsequent views are in some sense reactions to those elaborated in this paper. We will examine the main objections formulated in the 60s and the 70s, and how they ultimately led to three main alternative views, namely, Salmon's causal-mechanical account, Kitcher and Friedman's unification approach, and van Fraassen's pragmatic account. From there, we will examine some secondary literature that will help us to understand how research agendas were set at the turn of the 21st century. Our course will conclude with recent discussions of idealization and asymptotic explanation in the mathematical sciences.