The elastodynamic problem is formulated for massive and lumped elasticity systems, solicited by time-dependent forces and/or base motions. Inertia forces and mass matrices are introduced according to the D’Alembert Principle. As alternatives to the direct formulation, the Virtual Work and the Energy Formulations (this latter expressed by the Hamilton Principle), are illustrated. The free motions of the system are studied first, leading to the evaluation of natural frequencies and natural modes, characterized by certain orthogonal properties. Elastically or kinetically semi-definite systems (i.e., under-constrained or lacking-mass systems) are also analyzed, requiring static condensation or revealing the existence of floppy modes. The influence of a state of pre-stress on the modal properties of the system is discussed. The response to harmonically varying forces is successively addressed. The notion of dynamic amplification factor is introduced and the phenomenon of resonance is highlighted. Base-excited systems are considered, too. Then, the Stability of Equilibrium Problem is stated in the dynamic context, according to the Lyapunov definition. The analysis is carried out linearizing the motion around a known equilibrium point, as governed by the variational equation. Elastic systems loaded by conservative or non-conservative forces are studied, by distinguishing these latter in position-dependent (circulatory) and velocity-dependent forces. In all cases, general criteria for assessing stability, based on the type of the characteristic exponents, are given.