The fabrication of superhydrophobic-oleophobic HfO₂ coatings presents a challenge. Here, we synthesize via the self-organized anodizing of aluminum-on-hafnium layers sputter-deposited onto SiO₂ and glass substrate arrays of highly aligned HfO₂ nanostructures of dissimilar shapes, sizes, spacings, and population densities termed as nanorods, nanopillars, nanohoodoos, and nanopillars-in-domains, which were then modified with various self-assembled monolayers. The treatment in fluoroalkyl-trimethoxysilane (FAS-17) appeared to be the most effective in making the hafnium-oxide nanoarrays superhydrophobic, with water contact angles of 153° (nanopillars), 155° (nanorods), 160° (nanohoodoos), and 174° (nanopillars-in-domains), the latter revealing the smallest roll-off angle of <1°. The FAS-17-modified HfO₂ nanorods and pillars-in-domains demonstrated the best oleophobic properties with a contact angle for ethylene glycol of 140° and for rapeseed oil of 121°. In addition, the HfO₂ nanorod films were highly transparent and antireflective in the visible spectral range and substantially less transparent and increasingly reflective in the near infrared. An outstanding combination of robust superhydrophobic-oleophobic properties with the specific optical behavior of the HfO₂ nanoarrays makes them attractive for application as self-cleaning visible-transparent heat-repelling dielectric coatings for solar cell coverglass and concentrator photovoltaics.