As noted by the author, previous work in several papers motivates this article. Perhaps principal among them is a paper by F. Forstnerič and J. Winkelmann [Math. Res. Lett. {\bf 12} (2005), no.~2-3, 265--268; MR2150882], in which it was shown that the set of all analytic discs on $\Delta$ with dense image in a complex manifold $Y$ is dense in the set of all analytic discs $\Delta \to Y$, with respect to the compact open topology. Four approximation theorems for $Y$-valued functions are given. Among them is the following (Theorem 1.2): Let $X$ be a connected manifold and let $Y$ be a Stein manifold. Denote by $\Cal H(X,Y)$ the set of all holomorphic maps with dense image in $Y$, and by $\Cal G(X,Y)$ the set of all holomorphic maps that are non-constant and have relatively compact image in $Y$. Then for every compact set $M \subset X$, every $\varepsilon > 0$, and every $g \in \Cal G(X,Y)$, there is $h \in \Cal H(X,Y)$ such that $\sup _{x\in M} d_Y(g(x),h(x)) < \varepsilon$. The proof of this and the other three results make fundamental use of a modified version of Theorem 1.1 of F.~S. Deng, J.~E. Fornæss and E.~F. Wold [Proc. Amer. Math. Soc. {\bf 146} (2018), no.~6, 2473--2487; MR3778150].