The control of dispersion in fibre optical waveguides is of critical importance to optical fibre communications systems(1,2) and more recently for continuum generation from the ultraviolet to the mid-infrared(3-5). The wavelength at which the group velocity dispersion crosses zero can be set by varying the fibre core diameter or index step(2,6-8). Moreover, sophisticated methods to manipulate higher-order dispersion so as to shape and even flatten the dispersion over wide bandwidths are possible using multi-cladding fibres(9-11). Here we introduce design and fabrication techniques that allow analogous dispersion control in chip-integrated optical microresonators, and thereby demonstrate higher-order, wide-bandwidth dispersion control over an octave of spectrum. Importantly, the fabrication method we employ for dispersion control simultaneously permits optical Q factors above 100 million, which is critical for the efficient operation of nonlinear optical oscillators. Dispersion control in high-Q systems has become of great importance in recent years with increased interest in chip-integrable optical frequency combs(12-32).