The vulnerability of humankind to SARS-CoV-2 in the absence of a pre-existing immunity, the unpredictability of the infection outcome, and the high transmissibility, broad tissue tropism, and ability to exploit and subvert the immune response pose a major challenge and are likely perpetuating the COVID-19 pandemic. Nevertheless, this peculiar infectious scenario provides researchers with a unique opportunity for studying, with the latest immunological techniques and understandings, the immune response in SARS-CoV-2 naive versus recovered subjects as well as in SARS-CoV-2 vaccinees. Interestingly, the current understanding of COVID-19 indicates that the combined action of innate immune cells, cytokines, and chemokines fine-tunes the outcome of SARS-CoV-2 infection and the related immunopathogenesis. Indeed, the emerging picture clearly shows that the excessive inflammatory response against this virus is among the main causes of disease severity in COVID-19 patients. In this review, the innate immune response to SARS-CoV-2 infection is described not only in light of its capacity to influence the adaptive immune response towards a protective phenotype but also with the intent to point out the multiple strategies exploited by SARS-CoV-2 to antagonize host antiviral response and, finally, to outline inborn errors predisposing individuals to COVID-19 disease severity.