Since the seminal work of Gidas, Ni & Nirenberg [Math. Anal. Appl. Part A, 1981], considerable effort has been devoted to the classification of positive solutions to the critical $p$-Laplace equation and, more broadly, to the study of symmetry and monotonicity properties of positive solutions to equations modeled on the critical one. Following Struwe’s qualitative stability result [Math. Z., 1984], several works have addressed quantitative aspects of the stability of the critical equation in the case $p=2$ and for functions in the energy space $\mathcal{D}^{1,2}(\mathbb R^n)$. In this seminar, we present recent stability results for the general case $1 < p < n$, as well as quasi-symmetry and monotonicity properties of both energy and non-energy solutions for a broader class of semilinear equations. Time permitting, we will also discuss further results concerning the classification of local weak solutions to the critical $p$-Laplace equation. These are joint works with Giulio Ciraolo and Matteo Cozzi (Università degli Studi di Milano).