  Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is
known that $L\subset M$ and that $M\setminus L\neq\varnothing$. In this work,
we exhibit new gaps of $L$ and $M$ using two methods. First, we derive such
gaps by describing a new portion of $M\setminus L$ near to 3.938: this region
(together with three other candidates) was found by investigating the pictures
of $L$ recently produced by V. Delecroix and the last two authors with the aid
of an algorithm explained in one of the appendices to this paper. As a
by-product, we also get the largest known elements of $M\setminus L$ and we
improve upon a lower bound on the Hausdorff dimension of $M\setminus L$
obtained by the last two authors together with M. Pollicott and P. Vytnova
(heuristically, we get a new lower bound of $0.593$ on the dimension of
$M\setminus L$). Secondly, we use a renormalisation idea and a thickness
criterion (reminiscent from the third author's PhD thesis) to detect infinitely
many maximal gaps of $M$ accumulating to Freiman's gap preceding the so-called
Hall's ray $[4.52782956616...,\infty)\subset L$.
