In the first part of this talk we survey various results related to Loewner chains, the generalized Loewner differential equation, and Herglotz vector fields on the Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. Extremal problems and related results for the family $S_A^0(\mathbb{B}^n)$ of univalent mappings with $A$-parametric representation on $\mathbb{B}^n$ will be also discussed, where $A\in L(\mathbb{C}^n)$ such that $k_+(A)< 2m(A)$. Here $k_+(A)$ is the Lyapunov index of the operator $A$ and  $m(A)=\min_{\|z\|=1}\Re\langle A(z),z\rangle$. Next, we present recent results on approximation properties of various families of normalized univalent mappings $f$ on $\mathbb{B}^n$, with Runge image, by automorphisms of $\mathbb{C}^n$ and smooth quasiconformal diffeomorphisms of $\mathbb{C}^n$ onto itself, whose restrictions to $\mathbb{B}^n$ have the same geometric property as the initial mappings $f$. Open problems and conjectures will be also considered.  \bigskip  Joint work with Ian Graham (Toronto), Hidetaka Hamada (Fukuoka), Gabriela Kohr (Cluj-Napoca), and Mihai Iancu (Cluj-Napoca).