This is the first of three colloquium lectures given by the author at the annual AMS meeting 1997. It emphasizes the role of intrinsic volumes as a basic notion in geometric probability. After passing from volume and surface area to mean width (in 3-space), the Euler characteristic is discussed in more detail. Integral geometric representations are mentioned and used to extend the intrinsic volumes additively to unions of convex sets. The lecture culminates in Hadwiger's celebrated characterization theorem: Any continuous, motion invariant additive functional on convex bodies is a linear combination of intrinsic volumes.