A high-efficiency, Multiple-Beam Klystron (MBK), designated the VKL-8301, is being manufactured for the DESY Tera Electron volt Superconducting Linear Accelerator (TESLA) in Hamburg, Germany. There are a number of excellent reasons for using an MBK for this application. The primary reasons are reduced size and lower operating voltage with respect to the conventional, single beam counterparts. Once this decision has been made, the class of MBK must now be selected. MBKs can be divided into two categories: Fundamental Mode (FM) and Higher-order Mode (HM) devices, distinguished by the interaction mode of the cavity resonators. Each class has inherent advantages and disadvantages dependent upon end-user requirements. For the 10 MW, 1.3 GHz TESLA application the HM-MBK is the clear choice. The primary factor influencing this choice was operational life, since the accelerator will require approximately 600 MBKs. The advantage of the HM approach is low cathode loading. Our cathode loading design goal of 2 A/cm/sup 2/ or less has been achieved. For this application the HM-MBK cathode loading is a factor of four lower than competing FM-MBK designs and a factor of three lower than the SLAC 5045 design. The VKL-8301 will use six off-axis electron beams interacting with a combination of TM/sub 010/ and hybrid TM/sub 020/ cavities. These six beams are equally spaced on a diameter of approximately 25 centimeters. Because of the large beam-to-beam separation, individual high-area convergence guns can be utilized versus the single multi-emission-site gun used in FM-MBK's. This solution requires a sophisticated focusing system that is relatively difficult to realize, compounded by our use of confined-flow focusing. Newly developed, state-of-the-art three- dimensional electromagnetics codes have been used to design the novel electron-beam-focusing system and microwave cavity geometry. Modeling and simulation results will be presented, hardware will be shown, and a description of the FM- versus HM-MBK selection process will be discussed.