The binding of T cell immune checkpoint proteins programmed death 1 (PD-1) and cytotoxic T-lymphocyte-associated protein 4 (CTLA-4) to their ligands allows immune evasion by tumours. The development of therapeutic antibodies, termed checkpoint inhibitors, that bind these molecules or their ligands, has provided a means to release this brake on the host anti-tumour immune response. However, these drugs are costly, are associated with potentially severe side effects, and only benefit a small subset of patients. It is therefore important to identify biomarkers that discriminate between responders and non-responders. This review discusses the determinants for a successful response to antibodies that bind PD-1 or its ligand PD-L1, dividing them into markers found in the tumour biopsy and those in non-tumour samples. It provides an update on the established predictive biomarkers (tumour PD-L1 expression, tumour mismatch repair deficiency and tumour mutational burden) and assesses the evidence for new potential biomarkers.