Cavity-enhanced methods have been extended to fiber optics by use of fiber Bragg gratings (FBGs) as reflectors. High-finesse fiber cavities were fabricated from FBGs made in both germanium/boron-co-doped photosensitive fiber and hydrogen-loaded Corning SMF-28 fiber. Optical losses in these cavities were determined from the measured Fabry-Perot transmission spectra and cavity ring-down spectroscopy. For a 10-m-long single-mode fiber cavity, ring-down times in excess of 2 ms were observed at 1563.6 nm, and individual laser pulses were resolved. An evanescent-wave access block was produced within a fiber cavity, and an enhanced sensitivity to optical loss was observed as the external medium's refractive index was altered.