The basic geometric structure in h-projective geometry is the family of h-planar curves, associated to a given Kähler metric. Such curves can be seen as generalisations of geodesics on Kähler manifolds. In this context, one problem of interest is the investigation of Kähler manifolds admitting another Kähler metric having the same h-planar curves as the given one. Such a pair of metrics is called h-projectively equivalent. Besides a general introduction to h-projective geometry I want to present a result which was obtained in a joint work with V. S. Matveev: every compact Kähler manifold which admits an h-projective vector field (that is a one-parameter group of transformations mapping the metric to an h-projectively equivalent one) is isomorphic to the complex projective space with Fubini-Study metric provided the h-projective vector field is not a Killing vector field.