This paper contains a number of interesting results on proper holomorphic mappings from a strictly pseudoconvex domain D to a (higher-dimensional) ball \({\mathbb{B}}^ N\). The first result is that there are domains D with smooth real-analytic boundary such that no proper mapping \(f: D\to {\mathbb{B}}^ n\) extends smoothly to \(\bar D.\) (A similar result has also been obtained by J. Faran.) The next result is a strengthening of an embedding theorem of Fornaess and Henkin: D can be nicely embedded into a bounded strictly convex domain with real analytic boundary. The third result is that for any holomorphic mapping \(h=(h_ 1,...,h_ p): D\to {\mathbb{C}}^ p\) with \(h\in C(\bar D)\) and \(h(\bar D)\subset {\mathbb{B}}^ p\), there are holomorphic functions \(f_ 1,...,f_ s\) such that \((h_ 1,...,h_ p,f_ 1,...,f_ s)\) maps D properly to \({\mathbb{B}}^{p+s}\). (A similar result has also been obtained by E. Low.)