In this paper, we have studied about design and fabrication of the actively shielded superconducting MRI magnet. Nonlinear optimization methods are usually used to find optimum coil configurations. However the selection of initial coil configurations is very difficult. In case bad initial data are used, it is even impossible to find optimum coil configurations which satisfy predefined constraints. We have developed computer optimization program which consists of two steps. Initial coil configurations are easily selected through linear optimization in the first step and optimum coil configurations are found through nonlinear optimization in the second step. We have also studied about superconducting shim coils to cancel error fields caused by coil fabrication errors. Many researchers published design concepts of shim coil. However all these studies are for shim coil design using filamentary coils with single turn, Shim coils with multi-turns should be used to produce enough field strength to cancel error fields. We have developed computer program for the design of shim coils which have proper thickness and length. An actively shielded superconducting MRI magnet with a small warm bore was fabricated and four sets of superconducting shim coils were equipped. The magnetic field distributions were measured and field correction was carried out using shim coils.