  We present a general shape optimisation framework based on the method of
mappings in the $W^{1,\infty}$ topology. We propose steepest descent and
Newton-like minimisation algorithms for the numerical solution of the
respective shape optimisation problems. Our work is built upon previous work of
the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), where a
$W^{1,\infty}$ framework for star-shaped domains is proposed. To illustrate our
approach we present a selection of PDE constrained shape optimisation problems
and compare our findings to results from so far classical Hilbert space methods
and recent $p$-approximations.
