Ionic conductivities in solid oxide fuel cell (SOFC) electrolytes yttria-stabilised zirconia (YSZ), calcia-stabilised zirconia (CSZ), gadolinium-doped ceria (GDC) and samarium-doped ceria (SDC) and cathode material lanthanum strontium cobalt oxide (LSCO) are directly calculated using DL AKMC, an adaptive kinetic Monte Carlo (aKMC) program which assumes limited a priori knowledge of the kinetics of systems. The materials were simulated over several milliseconds and over the range of experimentally most relevant temperatures and dopant concentrations (2-18 mol% for doped zirconia, 5-25 mol% for doped ceria and 5-80 mol% for LSCO). Ionic conductivities of the electrolytes at 1000 K are in good agreement with the observed values: CSZ in the range 3×10 −3 -1×10 −2 S cm −1 depending on dopant concentration, YSZ 4×10 −3 -3×10 −2 S cm −1 , GDC 1×10 −2 -5×10 −2 S cm −1 , SDC 1×10 −2 -7×10 −2 S cm −1 . LSCO is predicted to have an ionic conductivity of the order of 10 −2 -10 −1 S cm −1 depending on Sr content. Average activation energies over all migration processes are between 0.4-0.5 eV for the stabilised zirconias and between 0.2 and 0.3 eV for the doped cerias and 0.3 eV for LSCO, in agreement with experiment. aKMC provides a distinct advantage over traditional KMC methods, for which one has to provide a list of system state transitions. Here, all of the state transitions are dynamically generated, leading to a more accurate simulation of the kinetics as the system evolves.