The first part of the paper is devoted to algebras on one-dimensional varieties in $\mathsf{C}^n$ that are bounded by finite unions of mutually disjoint rectifiable simple closed curves. The relevant Shilov boundaries are considered, and certain nonapproximation phenomena are exhibited. The second part of the paper is devoted to the study of uniform algebras whose maximal ideal spaces are smooth surfaces and that admit sets of smooth generators. Such algebras are shown to consist of functions holomorphic off their Shilov boundaries.