Reflection groups, such as the symmetries of a kaleidoscope, are generated by reflections and are often finite. If we remove the mirrors and take the fundamental group of what is left, we get braid groups: infinite groups that, for example, record how the roots of a polynomial move around. After a quick look at fundamental groups acting on spaces, I will compare the relations in these two kinds of groups and see how they connect to geometry. Along the way, we will get a feel for the kinds of questions geometric group theorists ask about such groups, and why they ask them. The talk is aimed at first-year graduate students; the only prerequisite is the fundamental group.