Cellular Automata (CA) is an emerging paradigm for the combined analysis and design of complex systems using local update rules. An implementation of the paradigm has recently been demonstrated successfully for the design of truss and beam structures. In the present paper, CA is applied to two-dimensional continuum topology optimization problems. The topology problem is regularized using the popular SIMP approach. The design rules are derived based on continuous optimality criteria interpreted as local Kuhn-Tucker conditions. Both CA based and conventional finite element analyses are considered. Numerical experiments with the proposed algorithm indicate that the approach is quite robust and does not suffer from checkerboard patterns, mesh-dependent topologies, or numerical instabilities.