Harmonic map equations are an elliptic PDE system arising from the

minimisation problem of Dirichlet energies between two manifolds. In

this talk we present some some recent works concerning the symmetry

and stability of harmonic maps. We construct a new family of

''twisting'' examples of harmonic maps and discuss the existence,

uniqueness and regularity issues. In particular, we characterise of

singularities of minimising general axially symmetric harmonic maps,

and construct non-minimising general axially symmetric harmonic maps

with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.

Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$

to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for

$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and

Mazowiecka--Strzelecki.

(Joint work with Prof. Robert M. Hardt.)