The ffamiltonian problem is to determine whether a graph contains a spanning (Hamiltonian) path or cycle. Here we study the Hamiltonian problem for the generalized Fibonacci cubes, which are a new family of graphs that have applications in interconnection topologies [J. Liuand W.-J. Hsu, "Distributed Algorithms for Shortest-Path, Deadlock-Free Routing and Broadcasting in a Class of Interconnection Topologies," International Parallel Processing Symposium (1 99211. We show that each member of this family contains a Hamiltonian path. Furthermore, we also characterize the members of this family that contain a Hamiltonian cycle.