A novel class of circular resonators, based on a radial defect surrounded by Bragg reflectors, is studied in detail. Simple rules for the design and analysis of such structures are derived using a transfer matrix formalism. Unlike conventional ring resonators, annular Bragg resonators (ABR) are not limited by the total internal reflection condition and can exhibit both large free spectral ranges and low bend losses. The Bragg reflection mechanism enables the confinement of light within a defect consisting of a low refractive index medium (such as air). Strong atom-photon interaction can be achieved in such a structure, making it a promising candidate for sensing and cavity quantum electrodynamics applications. For sensing applications, we show that the ABR structure can possess significantly higher sensitivity when compared to a conventional ring resonator sensor. Lasing action and low threshold levels are demonstrated in ABR lasers at telecommunication wavelengths under pulsed optical pumping at room temperatures. The impact of the intensity and dimensions of the pump spot on the emitted spectrum is studied in detail.