Suppose that, for a surface $S\subset\bR^n$ with (weighted) surface measure $\sigma$ and for some $p,q$ with $p\in(1,2)$, the Fourier restriction operator $\cR:f\longmapsto \widehat f_S$ satisfies the inequality $$ \|\mathcal R f\|_{L^q(S,\sigma)}\le C\|f\|_{L^p(\mathbb R^n)}\ ,\qquad \forall f\in\cS(\mathbb R^n)\ . $$ Then extendability of $\mathcal R$ to all of $L^p(\mathbb R^n)$ indicates, heuristically, that, for general $f\in L^p(\mathbb R^n)$, $\widehat f$ can be assigned values on $S$, despite the fact that it is only defined a.e. The notion of ``maximal restriction operator'' has been introduced in a paper of 2019 by D.~M\"uller, J.~Wright and myself, for the purpose of giving measure-theoretic ground to this statement. In this talk I give a precise presentation of the problem, the improvements of our original result by various authors and some of the open problems.