Chimeric antigen receptor (CAR) T-cell therapies and bispecific T-cell engagers have produced durable remissions in haematological malignancies that were previously considered refractory, and the bispecific class has now extended this approach to a solid tumour, extensive-stage small-cell lung cancer, through the delta-like ligand 3 engager tarlatamab. The shared mechanism of intense, sustained T-cell activation generates two characteristic and potentially fatal toxicities: cytokine release syndrome (CRS) and immune effector cell-associated neurotoxicity syndrome (ICANS). As bispecific agents shift toward subcutaneous and outpatient administration, an increasing proportion of treated patients present to emergency departments without immediate access to the treating cellular therapy centre, where the syndrome may not be recognised. This narrative review translates the consensus produced by haematology and cellular therapy societies into an operational framework for acute care. It synthesises the mechanism, epidemiology, grading, and initial management of CRS, ICANS, and related immune effector cell toxicities, drawing on the American Society of Transplantation and Cellular Therapy consensus, the American Society of Clinical Oncology guideline, the joint European recommendations, and pivotal and updated product trials. Recognition rests on a few disciplined habits: naming the therapy, treating fever in a recently treated patient as an emergency, grading CRS and ICANS while methodically excluding infection, and contacting the treating centre early. Management centres on tocilizumab for CRS, corticosteroids for ICANS, anakinra in refractory disease, and aggressive supportive care. Competence with these syndromes is becoming a general expectation of acute care rather than the exclusive concern of cellular therapy units.