An extension of the classic thermodynamic theory to nanometer scale has generated a new interdisciplinary theory - nanothermodynamics. It is the critical tool for the investigation of the size-dependent physicochemical properties in nanocrystals. A simple and unified nanothermodynamic model for the melting temperature of nanocrystals has been established based on Lindemann’s criterion for the melting, Mott’s expression for the vibrational melting entropy, and Shi’s model for the size dependence of the melting point. The developed model has been extensively verified in calculating a variety of size- and dimensionality-dependent phase transition functions of nanocrystals. In this work, such a model was extended to explain the underlying mechanism behind the bandgap energy enhancement and Raman red shifts in semiconductor nanocrystals by (1) investigating the crystal size r, dimensionality d, and constituent stoichiometry x dependences of bandgap energies Eg in semiconductor quantum dots (QDs) and quantum wires (QWs); and (2) revealing the origin of size effect on the Raman red shifts in low dimensional semiconductors by considering the thermal vibration of atoms. For Eg, it is found that: (1) Eg increases with a decreasing r for groups IV, III-V and II-VI semiconductors and the quantum confinement effect is pronounced when r becomes comparable to the exciton radius; (2) the ratio of Eg(r, d)QWs/Eg(r, d)QDs is size-dependent, where Eg(r, d) denotes the change in bandgap energy; (3) the crystallographic structure (i.e. zinc-blende and wurtzite) effect on Eg of III-V and II-VI semiconductor nanocrystals is limited; and (4) for both bulk and nanosized III-V and II-VI semiconductor alloys, the composition effects on Eg are substantial, having a common nonlinear (bowing) relationship. For the Raman red shifts, the lower limit of vibrational frequency was obtained by matching the calculation results of the shifts with the experimental data of Si, InP, CdSe, CdS0.65Se0.35, ZnO, CeO2, as well as SnO2 nanocrystals. It shows that: (1) the Raman frequency (r) decreases as r decreases in both narrow and wide bandgap semiconductors; (2) with the same r, the sequence of size effects on (r) from strong to weak is nanoparticles, nanowires, and thin films; and (3) the Raman red shift is caused by the size-induced phonon confinement effect and surface relaxation. These results are consistent with experimental findings and may provide new insights into the size, dimensionality, and composition effects on the optical properties of semiconductors as well as fundamental understanding of high-performance nanostructural semiconductors towards their applications in optoelectronic devices.