Influenza A virus (IAV) polymerase complexes function in the nucleus of infected cells, generating mRNAs that bear 5' caps and poly(A) tails, and which are exported to the cytoplasm and translated by host machinery. Host antiviral defences include mechanisms that detect the stress of virus infection and arrest cap-dependent mRNA translation, which normally results in the formation of cytoplasmic aggregates of translationally stalled mRNA-protein complexes known as stress granules (SGs). It remains unclear how IAV ensures preferential translation of viral gene products while evading stress-induced translation arrest. Here, we demonstrate that at early stages of infection both viral and host mRNAs are sensitive to drug-induced translation arrest and SG formation. By contrast, at later stages of infection, IAV becomes partially resistant to stress-induced translation arrest, thereby maintaining ongoing translation of viral gene products. To this end, the virus deploys multiple proteins that block stress-induced SG formation: 1) non-structural protein 1 (NS1) inactivates the antiviral double-stranded RNA (dsRNA)-activated kinase PKR, thereby preventing eIF2α phosphorylation and SG formation; 2) nucleoprotein (NP) inhibits SG formation without affecting eIF2α phosphorylation; 3) host-shutoff protein polymerase-acidic protein-X (PA-X) strongly inhibits SG formation concomitant with dramatic depletion of cytoplasmic poly(A) RNA and nuclear accumulation of poly(A)-binding protein. Recombinant viruses with disrupted PA-X host shutoff function fail to effectively inhibit stress-induced SG formation. The existence of three distinct mechanisms of IAV-mediated SG blockade reveals the magnitude of the threat of stress-induced translation arrest during viral replication.