Because of their large anisotropy and layered structure, high T(c) superconducting oxides exhibit a long-range magnetic interaction between the Abrikosov vortices over typically many vortex spacings. This means non-local elasticity which softens the vortex lattice and strongly enhances its thermal fluctuations and pinning-caused distortions. The small coherence length and therefore weak pinning potential facilitate thermally activated depinning of the flux lines, which leads to a finite resistivity when a magnetic field is applied. At low current densities thermal depinning corresponds to a finite diffusivity of the magnetic flux, whose exponential temperature dependence explains a large variety of experiments in a natural way without the assumption of a vortex lattice melting transition.