Stability is an important property of all systems, whether natural or engineered. The stability of a finite-dimensional system is determined entirely by the eigenvalues of the matrix A, but for infinite-dimensional systems is more complicated. Various definitions are possible. The two types of stability most appropriate for the study of linear DPS, exponential and asymptotic stability, are defined. Although equivalent for finite-dimensional systems, an infinite-dimensional system can be asymptotic but not exponentially stable. Also, the spectrum of an infinite-dimensional system can contain elements besides eigenvalues, and it is possible that the spectrum does not entirely determine stability. Checkable conditions for stability are provided. The significant but not straightforward case of vibrations is treated separately.