The Quadratic Assignment Problem (QAP) is an important combinatorial optimization problem with applications in many areas including logistics and manufacturing. QAP is known to be NP-hard, a computationally challenging problem, which requires the use of sophisticated heuristics in finding acceptable solutions for most real-world data sets. In this paper, we present GPU-accelerated implementations of a 2opt and a tabu search algorithm for solving the QAP. For both algorithms, we extract parallelism at multiple levels and implement novel code optimization techniques that fully utilize the GPU hardware. On a series of experiments on the well-known QAPLIB data sets, our solutions, on average run an order-of-magnitude faster than previous implementations and deliver up to a factor of 63 speedup on specific instances. The quality of the solutions produced by our implementations of 2opt and tabu is within 1.03% and 0.15% of the best known values. The experimental results also provide key insight into the performance characteristics of accelerated QAP solvers. In particular, the results reveal that both algorithmic choice and the shape of the input data sets are key factors in finding efficient implementations.