We study the classical nonlinear dynamics and the quantum vibrational energy eigenstates of the molecule HOCl. The classical vibrational dynamics, at energies below the HO+Cl dissociation energy, contains several saddle-center and period doubling bifurcations. The saddle-center bifurcations are shown to be due to a 2:1, and at higher energies a 3:1, nonlinear resonance between bend and stretch motions in various periodic orbits. The sequence of bifurcations takes the system from nearly integrable at low energies to almost completely chaotic at energies near the HO+Cl dissociation energy. At energies above dissociation we study the chaotic scattering of the Cl atom off the HO dimer. This scattering is governed by a homoclinic tangle formed by the stable and unstable manifolds of a parabolic periodic orbit at infinity. We construct the first three segments of the homoclinic tangle in phase space and use scattering functions to investigate its higher-order structure. For the quantum system we use a discrete variable representation to efficiently calculate the Hamiltonian matrix. We find 365 even and 357 odd parity eigenstates with energies below the dissociation energy. By plotting the eigenstates in configuration space we show that almost every quantum eigenstate can be associated with one or more of the classical periodic orbits. The classical bifurcations that give rise to new periodic orbits are manifest quantum mechanically through the sudden appearance of new classes of eigenstates. Despite the high degree of chaos in the classical dynamics at energies near the dissociation energy most quantum eigenstates remain highly ordered with recognizable nodal patterns. We use R-matrix theory together with a discrete variable representation to calculate quasibound states with energies above the dissociation energy. We find quasibound states with lifetimes ranging over 5 orders of magnitude. Using configuration space plots and Husimi distributions we show that the long-lived quasibound states are supported by unstable periodic orbits in the classical dynamics and medium-lived quasibound states are spread throughout the chaotic region of the classical phase space. Short-lived quasibound states show some similarity to unstable periodic orbits that stretch along the dissociation channel. / text