Investigations of MgB2 and Fe-based superconductors in recent years have revealed many unusual effects of multiband superconductivity but manifestations of anisotropic multiband effects in the critical current density Jc have not been addressed experimentally, mostly because of the difficulties to measure Jc along the c-axis. To investigate the effect of very different intrinsic anisotropies of sigma and pi electron bands in MgB2 on current transport, we grew epitaxial films with tilted c-axis

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... THETA 19.5), which enabled us to measure the components of Jc both along the ab-plane and the c-axis using magneto-optical and transport techniques. These measurements were combined with scanning and transmission electron microscopy, which revealed terraced steps on the surface of the c-axis tilted films. The measured field and temperature dependencies of the anisotropic Jc(H) show that Jc,L parallel to the terraced steps is higher than Jc,T perpendicular to the terraced steps, and Jc of thinner films (50 nm) obtained from transport experiments at 0.1 T reaches 10 of the depairing current density Jd in the ab plane, while magneto-optical imaging revealed much higher Jc at lower fields. To analyze the experimental data we developed a model of anisotropic vortex pinning which accounts for the observed behavior of Jc in the c-axis tilted films and suggests that the apparent anisotropy of Jc is affected by current pairbreaking effects in the weaker π band. Our results indicate that the out-of-plane current transport mediated by the π band could set the ultimate limit of Jc in MgB2 polycrystals.