  Each finite algebra $\mathbf A$ induces a lattice~$\mathbf L_{\mathbf A}$ via
the quasi-order~$\to$ on the finite members of the variety generated
by~$\mathbf A$, where $\mathbf B \to \mathbf C$ if there exists a homomorphism
from $\mathbf B$ to~$\mathbf C$. In this paper, we introduce the question:
`Which lattices arise as the homomorphism lattice $\mathbf L_{\mathbf A}$
induced by a finite algebra $\mathbf A$?' Our main result is that each finite
distributive lattice arises as~$\mathbf L_{\mathbf Q}$, for some quasi-primal
algebra~$\mathbf Q$. We also obtain representations of some other classes of
lattices as homomorphism lattices, including all finite partition lattices, all
finite subspace lattices and all lattices of the form $\mathbf L\oplus \mathbf
1$, where $\mathbf L$ is an interval in the subgroup lattice of a finite group.
