The coronavirus SARS-CoV-2 is the cause of the ongoing pandemic of COVID-191. Coronaviruses have developed a variety of mechanisms to repress host mRNA translation to allow the translation of viral mRNA, and concomitantly block the cellular innate immune response2,3. Although several different proteins of SARS-CoV-2 have previously been implicated in shutting off host expression4-7, a comprehensive picture of the effects of SARS-CoV-2 infection on cellular gene expression is lacking. Here we combine RNA sequencing, ribosome profiling and metabolic labelling of newly synthesized RNA to comprehensively define the mechanisms that are used by SARS-CoV-2 to shut off cellular protein synthesis. We show that infection leads to a global reduction in translation, but that viral transcripts are not preferentially translated. Instead, we find that infection leads to the accelerated degradation of cytosolic cellular mRNAs, which facilitates viral takeover of the mRNA pool in infected cells. We reveal that the translation of transcripts that are induced in response to infection (including innate immune genes) is impaired. We demonstrate this impairment is probably mediated by inhibition of nuclear mRNA export, which prevents newly transcribed cellular mRNA from accessing ribosomes. Overall, our results uncover a multipronged strategy that is used by SARS-CoV-2 to take over the translation machinery and to suppress host defences.