Focusing of waves in one dimension is analyzed for the plasma-wave equation and the wave equation with variable speed. The existence of focusing causal solutions to these equations is established, and such wave solutions are constructed explicitly by deriving an orthogonality relation for the time-independent Schrödinger equation. The connection between wave focusing and inverse scattering is studied. The potential at any point is recovered from the incident wave that leads to focusing to that point. It is shown that focusing waves satisfy certain temporal-antisymmetry and support properties. Discontinuities in the spatial and temporal derivatives of the focusing waves are examined and related to the discontinuities in the potential of the Schrödinger equation. The theory is illustrated with some explicit examples.