The interaction of the intense charged particles in a vacuum gap has remained an area of considerable interest to accelerator physics, sheath physics, high current diodes, high power microwave sources, vacuum micro- and nano-electronics. Previously, most of the developments of space-charge-limited (SCL) flows are focused on one-dimensional (1D) classical models and the multi-dimensional SCL flows (especially for 3D features) are relatively undeveloped. The classical Child-Langmuir law, which predicts the maximum space-charge-limiting current that can be transported across a 1D planar gap under steady-state condition, is one of the most important considerations used in various types of high current diodes, triodes and multi-stage acceleration gaps. This thesis presents recent advances in the theories of multi-dimensional SCL flows in various operating regimes such as quantum, short pulse and transition from field emission.