The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinite-dimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin-Voigt viscoelastic damping are presented.