Two basic physical models, a two-level system and a harmonic oscillator, are realized on the mesoscopic scale as coupled qubit and resonator. The realistic system includes moreovel the electronics for controlling the distance between the qubit energy levels and their populations and to read out the resonator's state, as well as the unavoidable dissipative environment. Such rich system is interesting both for the study of fundamental quantum phenomena on the mesoscopic scale and as a promising system for future electronic devices. We present recent results for the driven superconducting qubit - resonator system, where the resonator can be realized as an LC circuit or a nanomechanical resonator. Most of the results can be described by the semiclassical theory, where a qubit is treated as a quantum two-level system coupled to the classical driving field and the classical resonator. Application of this theory allows to describe many phenomena for the single and two coupled superconducting qubits, among which are the following the equilibrium-state and weak-driving spectroscopy, Sisyphus damping and amplification, Landau - Zener - Stuckelberg interferometry, the multiphoton transitions of both direct and ladder-type character, and creation of the inverse population for lasing.