Many parameters of non-equilibrium plasma generated by high intensity and fast lasers depend on the pulse intensity and the laser wavelength. In conditions favourable for the target normal sheath acceleration (TNSA) regime the ion acceleration from the rear side of the target can be enhanced by increasing the thin foil absorbance through the use of nanoparticles and nanostructures promoting the surface plasmon resonance effect. In conditions favourable for the backward plasma acceleration (BPA) regime, when thick targets are used, a special role is played by the laser focal position with respect to the target surface, a proper choice of which may result in induced self-focusing effects and non-linear acceleration enhancement. SiC detectors employed in the time-of-flight (TOF) configuration and a Thomson parabola spectrometer permit on-line diagnostics of the ion streams emitted at high kinetic energies. The target composition and geometry, apart from the laser parameters and to the irradiation conditions, allow further control of the plasma characteristics and can be varied by using advanced targets to reach the maximum ion acceleration. Measurements using advanced targets with enhanced the laser absorption effect in thin films are presented. Applications of accelerated ions in the field of ion source, hadrontherapy and nuclear physics are discussed.