  We study various generalisations of rationally connected varieties, allowing
the connecting curves to be of higher genus. The main focus will be on free
curves $f:C\to X$ with large unobstructed deformation space as originally
defined by Koll\'ar, but we also give definitions and basic properties of
varieties $X$ covered by a family of curves of a fixed genus $g$ so that
through any two general points of $X$ there passes the image of a curve in the
family. We prove that the existence of a free curve of genus $g \geq 1$ implies
the variety is rationally connected in characteristic zero and initiate a study
of the problem in positive characteristic.
