  A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for
every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism
$j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $\phi$ of $\cal S$
such that $\phi\circ j=f$. In this paper, we characterise the quasi-projective
posets and lattices of arbitrary cardinalities, finite permutations, graphs and
digraphs of arbitrary cardinalities with loops and without loops, finite
hypergraphs, and finite point-line geometries.
