Guided by a general framework for wavefront engineering, experiments demonstrate that in a light field, lines of zero intensity can be shaped into knotted and linked loops of arbitrary topology. Natural and artificially created light fields in three-dimensional space contain lines of zero intensity, known as optical vortices1,2,3. Here, we describe a scheme to create optical beams with isolated optical vortex loops in the forms of knots and links using algebraic topology. The required complex fields with fibred knots and links4 are constructed from abstract functions with braided zeros and the knot function is then embedded in a propagating light beam. We apply a numerical optimization algorithm to increase the contrast in light intensity, enabling us to observe several optical vortex knots. These knotted nodal lines, as singularities of the wave’s phase, determine the topology of the wave field in space, and should have analogues in other three-dimensional wave systems such as superfluids5 and Bose–Einstein condensates6,7.