This is an English translation of Ludwig Bieberbach's paper ``Remarks on Hilbert's Thirteenth Problem" originally written in German and originally published in Journal für die Reine und Angewandte Mathematik - 165 (89-92) 1931, along with an addendum to the paper published in 1933. Beiberbach studies under what conditions are there functions of three variables which can or cannot be obtained by combining or nesting functions of two variables. In the addendum, Bieberbach acknowledges a fatal error in a proof in the original article and connects Hilbert's thirteenth problem to a related problem in the theory of differential equations.