We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. We end with some conjectures and open problems.