HP Memristors are suspended nano-wires surrounded by a floating ring of titanium oxide insulator forming a coaxial line whose impedance/resistance is a function of the cross section of the oxide ring. Current causes the ring compress or stretch giving a memristance function of roughly a parabola which saturates when the oxide is fully stretched or compressed. Last I looked, there are a lot of legal battles currently ongoing and patent litigation involving HP and memristors which is tying the technology up.
Other Memristors commonly work by dendrite formation along electric fields but this method is pretty dang slow. Charge trapping semi-conductor junctions have been proposed but I haven't seen any myself.
This new memristor from U of M works by controlling Lithium diffusion via an electric field. The amount of Lithium available controls the formation of lattice networks in the substrate which in turn controls the conductivity of the material.