In this review we describe some new approaches in the surface photovoltage spectroscopy (SPS) performed in the metal-insulator-semiconductor operation mode, which are related to open questions in the SPS characterization of nanostructures. The first new approach is based on surface photovoltage (SPV) phase spectroscopy.  We show that i) the sign of the bandgap-related knee in the spectrum of the SPV phase modulus is positive (negative) for downward (upward) surface band bending and ii) for the case of non-linear recombination, the SPV phase spectrum reflects the features of the optical absorption coefficient spectrum. We demonstrate the necessity of simultaneous examination of both amplitude and phase SPV spectra in order to understand correctly the experimental data. As a second original approach, we have developed a vector model, in which the SPV signal is represented by a radial vector, containing all the information given by the amplitude and phase SPV spectra. Further, after a brief overview of the SPS characterization of nanostructures in the last decade, we present the results of our pioneering SPS studies of various nanostructures with graded interfaces including AlAs/GaAs superlattices with GaAs embedded quantum wells, multilayer InAs/InP quantum wires, InAs/InGaAlAs quantum dashes-in-well laser structures and multilayer InP/GaAs type II quantum dots. The interpretation of the amplitude and phase SPV spectra is carried our by the proposed vector model for the SPV signal in parallel with the analysis of data obtained by other experimental methods and electronic structure calculations. As a result valuable information about the investigated systems is obtained.