The behavior of power-law shot noise, for which the associated impulse response functions assume a decaying power-law form, is explored. Expressions are obtained for the moments, moment generating functions, amplitude probability density functions, autocorrelation functions, and power spectral densities for a variety of parameters of the process. For certain parameters the power spectral density exhibits 1/f-type behavior over a substantial range of frequencies, so that the process serves as a source of 1/f/sup alpha / shot noise for alpha in the range 0< alpha <2. For other parameters the amplitude probability density function is a Levy-stable random variable with dimension less than unity. This process then behaves as a fractal shot noise that does not converge to a Gaussian amplitude distribution as the driving rate increases without limit. Fractal shot noise is a stationary continuous-time process that is fundamentally different from fractional Brownian motion. Several physical processes that are well described by power-law noise in certain domains are considered: 1/f shot noise, Cherenkov radiation from a random stream of charged particles, diffusion of randomly injected concentration packets the electric field at the growing edge of a quantum wire, and the mass distribution of solid-particle aggregates.<>