In this thesis, from discrete spring-mass systems to continuous elastic solids, the possibility of achieving topological phases and elastic spin-Hall effect are analytically and numerically discussed. Originating from time-reversal symmetry breaking via applying external fields, a unidirectional and backscattering-immune edge state arises owing to the topological protection. Caused by the effective spin-orbit coupling, the elastic counterpart of spin-Hall effect arises at certain area of the momentum space. Also, the proposed arguments are verified by numerical calculation of practical mechanical crystals and elastic composites. We believe these studies pave the way for the future researches in topological elasticity. On the other hand, PT symmetry, which is a weaker restriction than Hermicity, allows real eigenvalues in a non-Hermitian Hamiltonian. However, it is challenging to introduce the PT condition into quantum mechanical systems. In this thesis, we consider an acoustic metamaterial made of periodically arranged spinning cylinders. By virtue of the rotational Doppler effects, the dispersion relation around the rotating speed of rods is significantly influenced by the rotation. The frequency shifts cause a PT symmetric Hamiltonian so that, at specific points, the spontaneous PT symmetry breaking emerges and exceptional points arise. Lastly a possible setup is discussed for the future experimental realisation.