In the town of Königsberg there were seven bridges connecting a number of islands. The question was whether a citizen could take a single walk that would cross every bridge exactly once. Euler solved this problem and a whole family of problems like this one, and we will discuss the solution that he found. We will also describe some of the  applications of this subject to contemporary DNA sequencing, so these questions are of practical importance as well as being very pretty. You might enjoy having a look at the web pages http://forum.swarthmore.edu/~isaac/problems/bridges1.html, and http://www.nature.com/nsu/010816/010816-9.html, to get a little background material. I'll also say a few words about how computers can now do some things that  used to be thought of as do-able only by human mathematicians. The list of such things is growing steadily.