Although strong magnetic fields cannot be conveniently "focused" like light, modern microfabrication techniques enable preparation of microstructures with which the field gradients - and resulting magnetic forces - can be localized to very small dimensions. This ability provides the foundation for magnetic tweezers which in their classical variant can address magnetic targets. More recently, the so-called negative magnetophoretic tweezers have also been developed which enable trapping and manipulations of completely nonmagnetic particles provided that they are suspended in a high-magnetic-susceptibility liquid. These two modes of magnetic tweezing are complimentary techniques tailorable for different types of applications. This Progress Report provides the theoretical basis for both modalities and illustrates their specific uses ranging from the manipulation of colloids in 2D and 3D, to trapping of living cells, control of cell function, experiments with single molecules, and more.