Most of the anticancer drugs shares this common mechanism which act by generating free radicals, like super oxide radicals etc. (for eg. Doxorubicin which undergoes quinone recycling and producing toxic oxidative free radicals and they can in turn produce peroxy nitrile like highly toxic products) which are highly toxic to the rapidly dividing cellular populations.
In due course of its action, they are not specific to cancer cells and can kill healthy normal rapidly dividing cells (gut mucosal lining cells, blood and hair cells etc.). Hence the side effects are unavoidable with chemotherapy, but at the same time they are life saving drugs when one considers Risk/Benefit analysis and also most of the side effects vanish over a period of time post chemotherapy.
On the other side of the free radical actions, they can also produce mutations in the tumor suppressor genes (which actually controls the cell division and prevents neoplastic disease) like P53 etc. which can lead to development of cancer. That is why most of the anticancer drugs can in turn produce neoplasia on prolonged treatment (for eg. Nitroso urea class).
Finally, generating free radicals against cancer is the most accepted mechanism of action for any class of anticancer drugs approved or in discovery pipe line.
I would like to add from other cancer modality called photodynamic therapy or PDT which also support the mechanism of action as explained by Dr, Grandi. Briefly, in PDT the energy transfer involves two pathways: Type I (radicals and reactive oxygen species) and Type II (singlet oxygen) these cascade eventually yield cytotoxic reactive oxygen species (ROS), which in a biological environment these toxic species can interact with cellular constituents causing biochemical disruption to the cell and trigger both apoptotic and necrotic for cancer cells. For more details here is useful source for you (Photodynamic therapy for cancer, Nature Reviews Cancer 3, 380-387 (May 2003)