Let A be the generator of a strongly continuous semigroup Ton the Hilbert space X, and let C be a linear operator from D(A) to another Hilbert space Y (possibly unbounded with respect to X. not necessarily admissible). We consider the problem of estimating the initial state z(0) is an element of D(A) (with respect to the norm of X) from the output function y(t) = CT(t)z(0), given for all t in a bounded interval [0, tau]. We introduce the concepts of estimatability and backward estimatability for (A, C) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z(0) from y. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell's principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrodinger equations, and we provide results from numerical simulations. (C) 2010 Elsevier Ltd. All rights reserved.