  We survey our contributions on the classification of elliptic fibrations on
K3 surfaces with a non-symplectic involution. We place them in the more general
framework of K3 surfaces with an involution without any hypothesis on its fixed
locus or on the action on the symplectic 2-form. We revisit the complete
classification of elliptic fibrations on K3 surfaces with a 2-elementary
N\'eron--Severi lattice, and give a complete classification of extremal
elliptic fibrations on K3 surfaces that are quadratic covers of rational
elliptic surfaces.
