We derive techniques for proving exact values or bounds on the meeting time. These techniques are used to prove two upper bounds on the meeting time of an arbitrary number of random walks in any connected undirected graph. Both bounds express the meeting time of several random walks in terms of the meeting times of fewer random walks. One of the bounds is tight on rings, the other is a generalization of a bound suggested by Tetali and Winkler. The techniques yield exact expressions for the meeting times on rings and for two or three random walks in hitting-time-symmetric graphs. Additional results include case study of various meeting time games on rings, several counterexamples to "natural" conjectures concerning meeting times, and partial characterization of hitting-time-symmetric graphs.