The dark matter problem is one of the most fascinating problems in modern astrophysics. Multiple lines of evidence point to its existence, yet, despite dedicated searches, no definite detection of it has yet been made. In this talk, I will look back at the historical development of the dark matter problem in astrophysics, starting with the work of Zwicky, and including the crucial developments in the 1970s when, by the end of that decade, most astrophysicists adopted the viewpoint that dark matter exists. Where possible, I will point out the present-day problems relevant to the presence of dark matter in our own Galaxy.