This talk concerns consistency of the estimators of $ell_{1}$ penalized high-dimensional regressions including linear regressions and generalized linear regressions. For linear regressions, we consider the linear models with weakly dependent errors, such as $alpha$-mixing, $rho$-mixing error sequences. We prove that the estimators obtained by lasso and square-root lasso are consistent even in the non-Gaussian error case. For generalized linear regressions, we propose a new penalized method to solve sparse Poisson regression problems. It can be viewed as penalized weighted score function method, which possesses a tuning-free feature. We show that under mild conditions, our estimator is $ell_{1}$ consistent and the tuning parameter can be pre-specified, which enjoys the same good property as the square-root lasso.