We will study the topology of braids, as well as how braids arise naturally in representation theory, dynamics, algorithms, combinatorics, algebraic functions, and counting points on varieties over finite fields. Our highlight is the intimate connection between braids and polynomials. We will explain several theorems which can be viewed under one common theme: "Problems about roots of polynomials are hard because the topology of braids is rich and complex." This course is particularly suitable for students who have taken a course in algebraic topology and wonder how they can use the theory to prove new theorems in and out of topology. Along the course, I will propose many open problems for students to think about.