The concept of broadband coherent supercontinuum (SC) generation in all-normal dispersion (ANDi) fibers in the near-infrared, visible and ultraviolet (UV) spectral regions is introduced and investigated in detail. In numerical studies, explicit design criteria are established for ANDi photonic crystal fiber (PCF) designs that allow the generation of flat and smooth ultrabroad spectral profiles without significant fine structure and with excellent stability and coherence properties. The key benefit of SC generation in ANDi fibers is the conservation of a single ultrashort pulse in the time domain with smooth and recompressible phase distribution. Self-phase modulation and optical wave breaking are identified as the dominant nonlinear effects responsible for the nonlinear spectral broadening. It is further demonstrated that coherence properties, spectral bandwidth and temporal compressibility are independent of input pulse duration for constant peak power. The numerical predictions are in excellent agreement with experimental results obtained in two realizations of ANDi PCF optimized for the near-infrared and visible spectral region. In these experiments, the broadest SC spectrum generated in the normal dispersion regime of an optical fiber to date is achieved. The exceptional temporal properties of the generated SC pulses are demonstrated by their application in ultrafast transient absorption spectroscopy and by their temporal recompression to sub-two cycle durations by linear chirp compensation. The concept is further extended to tapered optical fibers with submicron diameter. Realized either as freestanding photonic nanowire or suspended core fiber, these tapered ANDi fibers allow the generation of coherent SC spectra with considerable spectral power densities in the usually hard to reach wavelength region below 300 nm.