In globular star clusters, there can be about a million stars in a space roughly the size of the region encompassing the Sun and the nearest other star. In such regions, stars will occasionally make direct collisions with one another, and even more frequently pass by one another closely enough that they can produce tight binary systems. The direct collision products and the close binaries represent nearly all of the most unusual stars in the clusters. These include the "blue stragglers", which are stars that appear to be younger than the rest of the stars in the clusters and binary stars where a normal star passes matter to a neutron star or a black hole and produces strong X-ray emission. I will discuss the formation of these exotic stars.