Taylor instability involves five successive stages of mixing: infinitesimal amplitude, finite amplitude, Taylor's "asymptotic" interpenetration by spikes and roundended columns, breakup of these spikes, ant turbulent mixing. The existing theory of Taylor instability is critically reviewed and extended. This theory covers the first three stages in the growth of periodic disturbances of the interface separa- separating two non-viscous, incompressible fluids. (Using Fourier analysis, the first stage can be treated without assuming periodicity. ) The analysis is based on classical hydrodynamics, supplemented by modern numerical methods. It is indicated how the fourth stage can be treated similarly. However, the fifth stage must be treated by statistical methods, which also throw light on the first four stages. The application of statistical methods to Taylor instability will be treated in a later report. A supplement to the present report, consisting of appendices which give expanded mathematical discussions of special topics, will also appear later.