Projection of individual age-specific fertility rates is a forecasting problem of high dimension. We solve this dimensionality problem by using a scaled and shifted gamma curve to approximate the age-specific rates and forecasting the curve parameters using a multivariate time series model. The resulting time series forecasts of parameters are then used to project fertility curves, and hence individual age-specific fertility rates. This reduces the dimensionality of the forecasting problem and also guarantees that long run projections of age-specific fertility rates will exhibit a smooth shape across age similar to historical data. For short-term projections it is also important to forecast the age-specific deviations from the fitted curves, which can be done by simple methods.