In this paper, the authors provide an overarching analysis of primal-dual dynamics associated with linear equality-constrained optimization problems using contraction analysis. For the well-known standard version of the problem, they establish convergence under convexity and the contracting rate under strong convexity. Then, for a canonical distributed optimization problem, the authors use partial contractivity to establish global exponential convergence of its primal-dual dynamics. As an application, they propose a new distributed solver for least-squares problems with the same convergence guarantees. Finally, for time-varying versions of both centralized and distributed primal-dual dynamics, the authors exploit their contractive nature to establish bounds on their tracking error. To support the analyses, they introduce novel results on contraction theory.