An elastic Kπ partial-wave analysis is presented. It is based on high statistics data for the reactions K ± p → K ±π + n and K ± p → K ±π −Δ ++ at 13 GeV obtained in a spectrometer experiment performed at SLAC. For each reaction, a t-dependent parametrization of the production amplitudes provides information on both the Kπ mass dependence of the production mechanisms and on Kπ scattering. Knowledge of the t-dependence then allows a calculation of the Kπ partial-wave amplitudes for Kπ masses from 0.7 to 1.9 GeV. The results of such analyses using data for (i) the neutral recoil reactions, (ii) the Δ ++ recoil reactions, and (iii) both neutron and Δ ++ recoil reactions simultaneously, are presented. Besides the leading J P = 1 −, 2 +, and 3 − resonances at M K π = 0.896, 1.434, and 1.78 GeV, there is evidence in two of the four possible partial-wave solutions for a broad P-wave resonant-like structure in the region of 1700 MeV. The I = 1 2 S-wave magnitude rises slowly and smoothly to a maximum near 1400 MeV, but then decreases rapidly between 1400 and 1600 MeV. This structure is strongly indicative of an S-wave resonance near 1450 MeV. The charge-two Kπ reaction is dominated by S-wave scattering with a total cross section decreasing from 4 mb at 0.9 GeV to 2 mb at 1.5 GeV. Both the I = 1 2 S-wave below 1400 MeV and the I = 3 2 S-wave are well described by an effective range parametrization.