This paper deals with the robust guaranteed cost observer with guaranteed cost performance for a class of linear uncertain jump systems with state delay. The transition of the jumping parameters in systems is governed by a finite-state Markov process. Based on the stability theory in stochastic differential equations, a sufficient condition on the existence of the proposed robust guaranteed cost observer is derived. Robust guaran-teed cost observers are designed in terms of a set of linear coupled matrix inequalities. A convex optimization problem with LMI constraints is formulated to design the suboptimal guaranteed cost observers.