Abstract: We show that central limit theorems hold for high-dimensional normalized means hitting high-dimensional rectangles. These results apply even when p>> n. These theorems provide Gaussian distributional approximations that are not pivotal, but they can be consistently estimated via Gaussian multiplier methods and the empirical bootstrap. These results are useful for building confidence bands and for multiple testing via the step-down methods. Moreover, these results hold for approximately linear estimators. As an application we show that these central limit theorems apply to normalized Z-estimators of p> n target parameter in a class of problems, with estimating equations for each target parameter orthogonalized with respect to the nuisance functions being estimated via sparse methods. (This talk is based primarily on the joint work with Denis Chetverikov and Kengo Kato.)