What do flat surfaces like a cube or icosahedron have to do with billiards? The billiard question is simply: If you hit a billiard on a polygonally shaped billiard table and it continues indefinitely, will it eventually get near to every point on the table? The answer is fairly easy for rectangular shaped tables but more complicated for other shapes. In this talk we will discuss how flat surfaces arise from the discussion of billiards and look at some of the properties of flat surfaces, including a suitable interpretation of Euler's formula. This talk is the first of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk is a motivational introduction to flat surfaces and is intended for a general audience of Rose faculty and students. The second talk, to be given later in the year, will discuss additional concepts and problems about flat surfaces suitable for undergraduate research topics.