In this thesis, we study the relationship between the geometric shape of a region in the plane, and certain probabilistic information about the behavior of Brownian particles inside the region. The probabilistic information is contained in the function h(r), called the harmonic measure distribution function. Consider a domain Ω in the plane, and fix a basepoint z0. Imagine lining the boundary of this domain with fly paper and releasing a million fireflies at the basepoint z0. The fireflies wander around inside this domain randomly until they hit a wall and get stuck in the fly paper. What fraction of these fireflies are stuck within a distance r of their starting point z0? The answer is given by evaluating our h-function at this distance; that is, it is given by h(r). In more technical terms, the h-function gives the probability of a Brownian first particle hitting the boundary of the domain Ω within a radius r of the basepoint z0. This function is dependent on the shape of the domain Ω, the location of the basepoint z0, and the radius r. The big question to consider is: How much information does the h-function contain about the shape of the domain’s boundary? It is known that an h-function cannot uniquely determine a domain, but is it possible to construct a domain that generates a given hfunction? This is the question we try to answer. We begin by giving some examples of domains with their h-functions, and then some examples of sequences of converging domains whose corresponding h-functions also converge to the h-function. In a specific case, we prove that artichoke domains converge to the wedge domain, and their h-functions also converge. Using another class of approximating domains, circle domains, we outline a method for constructing bounded domains from possible hfunctions f(r). We prove some results about these domains, and we finish with a possible for a proof of the convergence of the sequence of domains constructed.