The three longest paths problem concerns whether every three longest paths in a connected graph intersect. This problem was first posed by Zamfirescu in 1991 and by Gallai at the 1995 British Combinatorial Conference. In this thesis, we investigate the problem for Apollonian networks. We begin by proposing a systematic method for generating Apollonian networks, which becomes essential to stating and proving the results in this thesis. Our main results advance one of two research directions. In the first, we address the problem for Apollonian networks by defining and investigating three subfamilies: X, Y , and Z. We prove that every three longest paths in a member of one of these subfamilies intersect. Moreover, for the subfamilies X and Z, we prove stronger results; namely, that every member of X is Hamiltonian and that every member of Y has a Gallai vertex. Our second research direction concerns the search for families of Apollonian networks with three longest paths that may not intersect. In Apollonian networks of small order, every three longest paths intersect by necessity, owing to the length of their longest paths. It was conjectured by Croucher that the Apollonian networks have longest paths of length such that every three will intersect. However, we refute this conjecture by constructing an infinite family of Apollonian networks with short longest paths. The Apollonian networks in this family are not only the first known Apollonian networks with this property but also the first known polyhedral graphs with it. We then prove that although these graphs have short longest paths, every such graph has a Gallai vertex, and therefore none can constitute a negative answer to the three longest paths problem.