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Continuation power flow is a powerful tool to simulate power system steady-state stationary behaviours with respect to a given power injection variation scenario. Although continuation power flow methods have been implemented in several commercial packages, they may be still too slow for online applications. The authors aim to improve the continuation power flow methods, mainly their speed and, to a less extent, their reliability. Nonlinear predictors are developed based on the polynomial interpolations. The authors' numerical studies show that continuation power flow with the proposed nonlinear predictors can be much faster than that with traditional linear predictors such as tangent or secant predictors. Of the nonlinear predictors, second-order polynomial approximation-based and third-order-based nonlinear predictors show their superior performance in speed. Continuation power flow with second-order nonlinear predictors is generally slightly faster than that with third-order nonlinear predictors. In addition, a hybrid corrector is developed and incorporated into continuation power flow. It is numerically shown on several test systems ranging from 118-bus to 1648-bus that continuation power flow with the proposed hybrid corrector can be much faster than that with traditional correctors such as the Newton method and the fast decoupled method. Finally, an improved continuation power flow with the developed nonlinear predictor and hybrid corrector is presented and evaluated.