This paper presents a Lagrangian approach for the simulation of two-dimensional free-surface flows along with a systematic validation. Fluid-flow is traditionally modeled using an Eulerian description in association with finite differences and, more recently, finite elements. Free-surface flow introduces new difficulties in connection with the time dependence of the domain. The proposed approach sets a framework for meshfree and finite element formulations; it is extended to two-phase media in Preisig and Zimmermann [M. Preisig, Th. Zimmermann, Two-phase free-surface fluid dynamics on moving domains, J. Comput. Phys. 229 (7) (2010)2740-2758]. In this paper triangular finite elements and a remeshing procedure to avoid excessive element distortion are chosen for their robustness. Convergence of the Lagrangian description is analyzed and compared to results obtained with an Eulerian description. The equivalence of the two formulations for confined flows is demonstrated numerically. Illustrations include stratified flow, flow over a backward-facing step, propagation of a solitary wave and dam break. Meshfree formulations will be addressed in separate papers. (C) 2010 Elsevier B.V. All rights reserved.