  We formulate a notion of stability for maps between polarised varieties which
generalises Kontsevich's definition when the domain is a curve and
Tian-Donaldson's definition of K-stability when the target is a point. We give
some examples, such as Kodaira embeddings and fibrations. We prove the
existence of a projective moduli space of canonically polarised stable maps,
generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions
one and two. We also state an analogue of the Yau-Tian-Donaldson conjecture in
this setting, relating stability of maps to the existence of certain canonical
K\"ahler metrics.
