Calculus of Variations in $L^\infty$ has a long history, the scalar case of which was initiated by G.Aronsson in the 1960s and is under active research ever since. Aronsson’s motivation to study this problem was related to the optimisation of Lipschitz Extensions of functions. Mathematically, minimising the supremum is very challenging because the equations are non-divergence and highly degenerate. However, it provides more realistic models, as opposed to the classical case of minimisation of the average (integral). However, due to fundamental difficulties, until the early 2010s the field was restricted to the scalar case. In this talk I will discuss the vectorial case, which has recently been initiated by the speaker. The analysis of the $L^\infty$-equations is based on a recently proposed general duality-free PDE theory of generalised solutions for fully nonlinear systems.