Based on the IsoDAR 60 MeV/amu compact cyclotron design, which introduces several novel ideas for this type of particle accelerator, we describe a new family of high-current cyclotrons with energy from 1 to 60 MeV for use as a tool for particle physics, isotope production, and other uses. The design is modular, with design features upstream of 1 MeV identical for any energy and easily customized equipment downstream to reach the desired energy. The cyclotrons accelerate 5 mA of H2+, which is then extracted and stripped of the binding electron to produce 10 mA of protons, regardless of design energy. Multiple beamlines can be fed from a single cyclotron by partially occluding the beam with extraction foils. Because the accelerators are low-cost and relatively small, they can be installed at local sites. We describe the main design features and novelties alongside an extensive simulation effort. We further describe several applications at 5 MeV (for 18F production for medical imaging), 15 MeV (nue flux measurements by targeting 27Al), and 60 MeV (IsoDAR, a definitive search for sterile neutrinos and non-standard interactions).