This computer simulation study carried out by classical the molecular dynamics technique (MD) investigates structural phase transitions of various atomic and molecular condensed matter systems. Particularly, our attention is focused on the signals for their observation. Classical nucleation theory is briefly reviewed in relation to the present study. An overall review is made on the methodology of the MD simulation technique such as the integration of the equations of motion, the rotational motion of molecules which is dealt with the four-component quaternion, and other various techniques involved with the simulation of atomic clusters, the simulation of bulk molecular systems using periodic boundary conditions, and isobaric molecular dynamics. There are also reviews on measurements of thermodynamic quantities which are monitored during the simulation including the pressure, the kinetic energy, and the potential energy as well as their corrections due to the spherical cutoff. In addition, various analysis techniques for the observation of the signals of structural phase transitions are discussed. All the potential functions used in this study are of the pairwise additive atom-atom Lennard-Jones interaction for both the atomic and the molecular systems. A small cluster of a binary mixture of krypton and argon atoms is studied by triggered breathing motions to investigate anharmonic motion which involves the structural phase transitions. A small cluster of krypton atoms is also simulated and discussed in detail concerning the very first moment of nucleation in relation to five-fold symmetric structures. In simulations of sulphurhexafluoride molecule systems, artifacts of finite size and periodic boundary conditions are investigated. Freezing the system by MD is investigated since it is known to supercool in computer simulations. Furthermore, accelerating the nucleation process by various methods such as shear flow, accelerated layer, inclusion of defect molecules, and pressure fluctuations is investigated.