The classical Brauer-Siegel theorem states that if $k$ runs through the sequence of normal extensions of $\mathbb{Q}$ such that $n_k/\log|D_k|\to 0,$ then $\log h_k R_k/\log \sqrt{|D_k|}\to 1.$ In this talk we will give a brief survey of what is known about the generalizations of this theorem. ... Third, we will dwell on the higher dimensional analogues of the Brauer-Siegel theorem both in the number field and in the function field cases giving an overview of known results and open problems. ...