On the basis of highly correlated ab initio calculations, an accurate determination of the electronic structure and of the rovibrational spectroscopy has been performed for the electronic ground state of the HZnF system. Using effective core pseudopotentials for the Zn and F atoms and associated aug-cc-pVQZ basis sets, we have calculated, at the multireference configuration interaction level including the Davidson correction, the three-dimensional potential energy surface of the X1Σ+ ground state. The rovibrational energy levels have been obtained variationally, and the results have been discussed and compared with existing experimental data on the ground state of the close system HZnCl, which exhibits a complicated vibration-rotation spectrum. Our analysis shows that the nature of the H-ZnF bond is quite similar to that of the H-ZnCl bond, according to their bond lengths, harmonic frequencies of the H-Zn stretching mode, and dissociation energies into H and ZnF/ZnCl. The ab initio study of the electronic ground and excited states of ZnH and ZnH+ are also presented using similar level of calculations. Characteristic constants are given for the first bounded electronic states correlating to the first two dissociation asymptotes of the neutral and ionic diatomics.