In this survey talk, I collect a lot of results from discrete and computational geometry, explaining the special role that "touchings" (tangencies) play in te subject. These questions contributed to the early development of the theory of Davenport-Schinzel sequences, algorithmic motion planning, geometric graph theory, and incidence geometry. We will also report on the recent solution of the Richter-Thomassen conjecture on intersecting closed (convex) curves.