We describe the design, construction, and performance of three generations of superconducting Ioffe magnetic traps. The first two are low current traps, built from four racetrack shaped quadrupole coils and two solenoid assemblies. Coils are wet wound with multifilament NbTi superconducting wires embedded in epoxy matrices. The magnet bore diameters are 51 and 105 mm with identical trap depths of 1.0 T at their operating currents and at 4.2 K. A third trap uses a high current accelerator-type quadrupole magnet and two low current solenoids. This trap has a bore diameter of 140 mm and tested trap depth of 2.8 T. Both low current traps show signs of excessive training. The high current hybrid trap, on the other hand, exhibits good training behavior and is amenable to quench protection.