Mean Field Annealing algorithm which was proposed for solving combinatorial opimization problems combines the properties of neural networks and Simulated Annealing. In this thesis, MFA is formulated for mapping problem in parallel processing and global rouing problem in physical design automation of Field Programmable Gate Arrays (FPGAs). A new Mean Field Annealing (MFA) formulation is proposed for the mapping problem for mesh-connected and hypercube architectures. The proposed MFA heuristic exploits the conventional routing scheme used in mesh and hypercube interconnection topologies to introduce an efficient encoding scheme. An efficient implementation scheme which decreases the complexity of the proposed algorithm by asymptotical factors is also developed. Experimental results also show that the proposed MFA heuristic approaches the speed performance of the fast Kernighan-Lin heuristic while approaching the solution quality of the powerful simulated annealing heuristic. Also, we propose an order-independent global routing algorithm for SRAM type FPGAs based on Mean Field Annealing The performance of the proposed global algorithm is evaluated in comparison with LocusRoute global router on ACM/SIGDA Design Automation benchmarks. Experimental results indicate that the proposed MFA heuristic peforms better than the LocusRoute.