  We study variational problems for second order supremal functionals $\mathrm
F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm
L^{\infty}(\Omega)}$, where $F$ satisfies certain natural assumptions, $\mathrm
A$ is a positive matrix, and $\Omega \Subset \mathbb R^n$. Higher order
problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$,
and exhibit a strikingly different behaviour compared to first order problems,
for which there exists an established theory, pioneered by Aronsson in 1960s.
The aim of this paper is to develop a complete theory for $\mathrm F_\infty$.
We prove that, under appropriate conditions, ``localised" minimisers can be
characterised as solutions to a nonlinear system of PDEs, which is different
from the corresponding Aronsson equation for $\mathrm F_\infty$; the latter is
only a necessary, but not a sufficient condition for minimality. We also
establish the existence and uniqueness of localised minimisers subject to
Dirichlet conditions on $\partial \Omega$, and also their partial regularity
outside a singular set of codimension one, which may be non-empty even if
$n=1$.
