We study classical solutions of SU(2) Yang-Mills field equations, with or without coupled scalar fields, in curved spacetimes. We consider, essentially, only static, spherically symmetric background metrics, for both Lorentz and Euclidean signatures. Section I presents the motivation for searching such solutions. In Sec. II, the equations of motion for a class of Ansätze and several static solutions are given, namely a class of scalar fields compatible with a point monopole and particular solutions, for gauge fields alone, in background Schwarzschild and de Sitter metrics, respectively. Some interesting properties are discussed. In Sec. III a finite-action, Prasad-Sommerfield-type solution is constructed for the O(4,1) de Sitter metric. In Sec. IV it is shown how one single, simple de Sitter solution can lead to various well-known flat-space solutions, and to new ones, through a systematic exploitation of conformal equivalence. Self-duality constraints are formulated explicitly in Sec. V for a static spherically symmetric metric. Certain results for the Robinson-Bertotti metric are given in Sec. VI.