In this paper we address the dead-beat estimation problem for the class of two-dimensional (2D) behaviors which are described by means of a 2D difference equation, and whose trajectories evolutions pertain the “nonnegative half-plane”. For this class of behaviors, the concept of nilpotency is defined and fully characterized. After this preliminary step, the estimation problem is formally addressed. Indeed, the concept of (consistent or not) 2D dead-beat observer (DBO) of the system relevant variables from the system measured variables is introduced, and a set of necessary and sufficient conditions for the existence of such a DBO is given. Finally, a complete parametrization of the family of all consistent DBOs is given, and some partial results, together with an interesting counterexample, regarding the class of (not necessarily consistent) DBOs are presented. To conclude the paper, the general dead-beat estimation theory developed in this paper is adjusted to address and solve several problems, like state estimation, the design of unknown input observers or the design of fault detectors and identifiers (possibly in the presence of disturbances), for 2D systems described by quarter-plane causal 2D state space models.