Abstract We study robust high-dimensional estimation of generalized linear models (GLMs); where a small number k of the n observations can be arbitrarily corrupted, and where the true parameter is high dimensional in the “p n” regime, but only has a small number s of non-zero entries. There has been some recent work connecting robustness and sparsity, in the context of linear regression with corrupted observations, by using an explicitly modeled outlier response vector that is assumed to be sparse. Interestingly, we show, in the GLM setting, such explicit outlier response modeling can be performed in two distinct ways. For each of these two approaches, we give `2 error bounds for parameter estimation for general values of the tuple (n, p, s, k).