In the same way that simple closed curves can be self-entangled in 3-space, 2-spheres can be self-entangled in 4-space, and 2-manifolds without boundary can exhibit entanglement complexity of various kinds. This talk will review some relatively recent work in this area. Most of the work to be described is joint work with Mahshid Atapour, Chris Soteros and De Witt Sumners.