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We study when there exists a dense holomorphic curve in a space ofholomorphic maps from a Stein space. We first show that for any bounded convexdomain $\Omega\Subset\mathbb{C}^n$ and any connected complex manifold $Y$, thespace $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc. Our secondresult states that $Y$ is an Oka manifold if and only if for any Stein space$X$ there exists a dense entire curve in every path component of$\mathcal{O}(X,Y)$.  In the second half of this paper, we apply the above results to the theory ofuniversal functions. It is proved that for any bounded convex domain$\Omega\Subset\mathbb{C}^n$, any fixed-point-free automorphism of $\Omega$ andany connected complex manifold $Y$, there exists a universal map $\Omega\to Y$.We also characterize Oka manifolds by the existence of universal maps.

Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps