Cancer therapy is aimed at the elimination of tumor cells and acts via the cessation of cell proliferation and induction of cell death. Many research publications discussing the mechanisms of anticancer drugs use the terms "cell death" and "apoptosis" interchangeably, given that apoptotic pathways are the most common components of the action of targeted and cytotoxic compounds. However, there is sound evidence suggesting that other mechanisms of drug-induced cell death, such as necroptosis, ferroptosis, autophagy, etc. may significantly contribute to the fate of cancer cells. Molecular cross-talks between apoptotic and nonapoptotic death pathways underlie the successes and the failures of therapeutic interventions. Here we discuss the nuances of the antitumor action of two groups of the widely used anticancer drugs, i.e., platinum salts and taxane derivatives. The available data suggest that intelligent interference with the choice of cell death pathways may open novel opportunities for cancer treatment.