This paper investigates the problems of two-dimensional (2-D) dissipative control and filtering for a linear discrete-time Roesser model. First, a novel sufficient condition is proposed such that the discrete-time Roesser system is asymptotically stable and 2-D (Q, S, R)-alpha-dissipative. Special cases, such as 2-D passivity performance and 2-D H-infinity performance, and feedback interconnected systems are also discussed. Based on this condition, new 2-D (Q, S, R)-alpha-dissipative state-feedback and output-feedback control problems are defined and solved for a discrete-time Roesser model. The design problems of 2-D (Q, S, R)-alpha-dissipative filters of observer form and general form are also considered using a linear matrix inequality (LMI) approach. Two examples are given to illustrate the effectiveness and potential of the proposed design techniques.