This is the first of three lectures given by Steven Strogatz in the spring of 2011 as part of the MIT Department of Mathematics' Simons Lecture Series. Copyrighted TV clips have been removed. The annual Simons Lecture Series celebrates the most exciting mathematical work by the very best mathematicians of our time. The format of this lecture series has evolved since its inception in 1999, and now includes two weeks of lectures - one in pure mathematics and the other in applied mathematics - given each spring. We are grateful to our good friend Jim Simons for providing the financial backing of these lectures.