When multiple light beams overlap in three-dimensional space, their interference produces tangled lines of complete darkness. These lines are called optical vortices and may be infinitely long, or form closed loops which can be linked or knotted. The vortex lines can be obtained from combining random waves (such as optical speckle). Alternatively, specific configurations of looped, linked or knotted vortex lines may be produced using holographic techniques to implement mathematically derived constructions. In the random superpositions, numerical experiments indicate that the tangle of vortex lines has a fractal nature, whereas the holographic approach allows the construction of isolated optical vortex knots. The presence of such a knot has implications for the topology of the whole field. Whether such topological features produced by interfering waves are merely curiosities or correspond to subtle physical phenomena remains an open question.