We forecast a single time series using many predictor variables with a new estimator called the three-pass regression filter (3PRF). It is calculated in closed form and conveniently represented as a set of ordinary least squares regressions. 3PRF forecasts converge to the infeasible best forecast when both the time dimension and cross section dimension become large. This requires only specifying the number of relevant factors driving the forecast target, regardless of the total number of common (and potentially irrelevant) factors driving the cross section of predictors. We derive inferential theory in the form of limiting distributions for estimated relevant factors, predictive coefficients and forecasts, and provide consistent standard error estimators. We explore two empirical applications that exemplify the many predictor problem: Forecasting macroeconomic aggregates with a large panel of economic indices, and forecasting stock market aggregates with many individual assets' price-dividend ratios. These, combined with a range of Monte Carlo experiments, demonstrate the 3PRF's forecasting power.