African swine fever virus is a complex double-stranded DNA virus that exhibits tropism for cells of the mononuclear phagocytic system. Virus replication is a multi-step process that involves the nucleus of the host cell as well the formation of large perinuclear sites where progeny virions are assembled prior to transport to, and budding through, the plasma membrane. Like many viruses, African swine fever virus reorganises the cellular architecture to facilitate its replication and has evolved multiple mechanisms to avoid the potential deleterious effects of host cell stress response pathways. However, how viral proteins and virus-induced structures trigger cellular stress pathways and manipulate the subsequent responses is still relatively poorly understood. African swine fever virus alters nuclear substructures, modulates autophagy, apoptosis and the endoplasmic reticulum stress response pathways. The viral genome encodes for at least 150 genes, of which approximately 70 are incorporated into the virion. Many of the non-structural genes have not been fully characterised and likely play a role in host range and modifying immune responses. As the field moves towards approaches that take a broader view of the effect of expression of individual African swine fever genes, we summarise how the different steps in virus replication interact with the host cell and the current state of knowledge on how it modulates the resulting stress responses.