The central solenoid is an intrinsic part of all present-day tokamaks and most spherical tokamaks. The spherical torus (ST) confinement concept is projected to operate at high toroidal beta and at a high fraction of the non-inductive bootstrap current as required for an efficient reactor system. The use of a conventional solenoid in a ST-based fusion nuclear facility is generally believed to not be a possibility. Solenoid-free plasma start-up is therefore an area of extensive worldwide research activity. Solenoid-free plasma start-up is also relevant to steady-state tokamak operation, as the central transformer coil of a conventional aspect ratio tokamak reactor would be located in a high radiation environment but would be needed only during the initial discharge initiation and current ramp-up phases. Solenoid-free operation also provides greater flexibility in the selection of the aspect ratio and simplifies the reactor design. Plasma start-up methods based on induction from external poloidal field coils, helicity injection and radio frequency current drive have all made substantial progress towards meeting this important need for the ST. Some of these systems will now undergo the final stages of test in a new generation of large STs, which are scheduled to begin operations during the next two years. This paper reviews research to date on methods for inducing the initial start-up current in STs without reliance on the conventional central solenoid.