Modern ion accelerators and ion implantation systems need very short, highly versatile, Low Energy Beam Transport (LEBT) systems. The need for reliable and continuous operation requires LEBT designs to be simple and robust. The energy efficiency of available high temperature superconductors (HTS), with efficient and simple cryocooler refrigeration, is an additional attraction. Innovative, compact LEBT systems based on solenoids designed and built with high-temperature superconductor will be developed using computer models and prototyped. The parameters will be chosen to make this type of LEBT useful in a variety of ion accelerators, ion implantation systems, cancer therapy synchrotrons, and research accelerators, including the ORNL SNS. The benefits of solenoids made with HTS will be evaluated with analytical and numerical calculations for a two-solenoid configuration, as will be used in the SNS prototype LEBT that will replace the electrostatic one at SNS, and a single solenoid configuration, as was proposed for the Fermilab proton driver that will be most applicable to ion implantation applications.