Quantum dots are small conducting islands that can often be usefully modeled as tiny capacitors. Though classical charging models can explain the Coulomb blockade of an isolated dot, they must be modified to explain the Coulomb blockade of dots coupled through the quantum mechanical tunneling of electrons. This thesis presents quantum mechanical models for pairs of tunnel-coupled dots and uses these models to follow the coupled dot blockade as it evolves from that characteristic of two isolated dots to that characteristic of a single composite dot. The primary aim is to find the relation between two quantities: the fractional peak splitting f and the dimensionless inter-DOT channel conductance g, both of which go from 0 to 1 as the isolated dot blockade changes into the composite dot blockade. The thesis begins with Chapter 1, which introduces coupled quantum dots and the Coulomb blockade and highlights the contents of the succeeding chapters. Chapters 2 and 3 present a transfer Hamiltonian model for weakly coupled dots and a one-dimensional backscattering model for strongly coupled dots. The leading and subleading terms in the weak and strong coupling expressions for f as a function of g are derived. The weak coupling calculation is performed via Rayleigh-Schroedinger perturbation theory about the endpoint (g, f) = (0,0). The strong coupling calculation employs the bosonization approach about the endpoint (g, f) = (1, 1). The results show substantial dependence on the number of inter-DOT tunneling channels N sub ch. Chapter 4 goes beyond the work of Chapters 2 and 3, which rely upon the assumption that tunneling and backscattering amplitudes can be treated as energy-independent.