Inhibition has long been thought to stabilize the activity of cortical networks at low rates, and to shape significantly their response to sensory inputs. In this talk, I will describe three recent collaborative projects that shed light on these issues. (1) I will show how optogenetic excitation of inhibition neurons is consistent with cortex being inhibition stabilized even in the absence of sensory inputs, and how this data can constrain the coupling strengths of E-I cortical network models. (2) Recent analysis of the effects of optogenetic excitation of pyramidal cells in V1 of mice and monkeys shows that in some cases this optogenetic input reshuffles the firing rates of neurons of the network, leaving the distribution of rates unaffected. I will show how this surprising effect can be reproduced in sufficiently strongly coupled E-I networks. (3) Another puzzle has been to understand the respective roles of different inhibitory subtypes in network stabilization. Recent data reveal a novel, state dependent, paradoxical effect of weakening AMPAR mediated synaptic currents onto SST cells. Mathematical analysis of a network model with multiple inhibitory cell types shows that this effect tells us in which conditions SST cells are required for network stabilization.