This thesis studies the physics of multilayer heterostructures grown from topological insulators (TIs), primarily bismuth selenide and antimony telluride, and other topologically trivial materials. This is done by extending a standard low-energy 3D TI Hamiltonian and varying its associated material parameters across the simulation domain. New results arising from the position-dependent TI interface model are found. For the first time, this method is incorporated into a density-functional theory (DFT) solver in order to study the self-consistent charge density in multilayer TI heterostructures due to the interface states. The thesis is structured as follows. The introduction (Ch. 1) presents a pedagogical review of the theory of 3D TIs and low-energy Hamiltonians used to study them, as well as typical methods in solid state physics that are made use of throughout the thesis. Chapter 2 presents the position-dependent Hamiltonian, showing new evidence for topological features of bulk states including varying degrees of band mixing and inversion; also, interface state tunnelling is shown to be affected by atomic layer orbital overlap, and incomplete localisation of surface states is demonstrated for antimony telluride. Chapter 3 presents a new DFT model of TI heterostructure interfaces and shows how conduction through TI interface states can be controlled with an electric field. Chapter 4 covers the extension of the model in Ch. 1 to 2D cross-sections of TI wires and heterostructures, showing for the first time evidence of localisation of conduction almost entirely within the inner interfaces of a 2D heterostructure wire. Chapter 5 presents our work with magnetic fields, demonstrating evolution of interface and bulk states with changing magnetic field and Landau level, as well as presenting new evidence for more complex spin structures in bismuth selenide arising from Landé factor signs. Our conclusions are presented in Chapter 6.