Given a domain in the complex plane and a basepoint in the domain, start a Brownian traveler at that basepoint. The harmonic measure distribution function, or $h$-function, of the domain gives information about where the Brownian traveler is likely to first hit the boundary of the domain. I will discuss methods of computing and approximating $h$-functions, give several examples, and present results about the convergence of sequences of $h$-functions.