Melnikov’s methods, which enable us to detect the existence of transverse homo- and heteroclinic orbits and periodic orbits, and their bifurcations, are now ones of classical techniques to study forced nonlinear oscillators. Recently, similar approaches have been developed to show the nonintegrability of such systems, which are not necessarily Hamiltonian. In this overview article, I explain the approaches to prove the nonintegrability of forced nonlinear oscillators. I begin with standard results of Melnikov’s methods for forced nonlinear oscillators, and give two definitions of integrability for Hamiltonian and more general systems. After briefly reviewing the classical work of Poincaré and Kozlov, the differential Galois theory and Morales-Ramis theory, I describe three new techniques to prove the nonintegrability of forced nonlinear oscillators and illustrate them for three types of the Duffing oscillators. Finally, I give some comments on related theoretical results and applications.