This paper explores the following question: what kind of statistical guarantees can be given when doing variable variable in high dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as ``screening'' and the last stage as ``cleaning.'' We consider three screening methods: the lasso, marginal regression, and forward stepwise regression. Our method also gives consistent variable selection under weak conditions.