The paper under review is a mini-monograph presenting in a systematic way recently published results (mostly by the author and his collaborators) related to various aspects of the theory of holomorphically invariant functions and pseudometrics. In particular, the author presents many new results and methods concerning the Lempert functions and Kobayashi metrics (Chapter 1), the holomorphic geometry of the symmetrized polydisc \(\mathbb G_n\) and the spectral ball \(\varOmega_n\) (Chapter 2), and estimates and boundary behavior of invariant metrics on \(\mathbb C\)-convex domains (Chapter 3).