Superhydrophobic surfaces (SHSs) and slippery lubricant-infused porous surfaces (SLIPSs) are receiving increasing attention for their excellent anti-icing, anti-fogging, self-cleaning and condensation heat transfer properties. The ability of such surfaces to passively shed and repel water is mainly due to the low-adhesion between the liquid and the solid surface, i.e., low contact angle hysteresis, when compared to hydrophilic or to hydrophobic surfaces. In this work we investigated the effect of surface structure on the condensation performance on SHSs and SLIPSs. Three different SHSs with structures varying from the micro- to the nano-scale were fabricated following easy and scalable etching and oxidation growth procedures. The condensation performance on such surfaces was evaluated by optical microscopy in a temperature and humidity controlled environmental chamber. On SHSs important differences on the size and on the number of the coalescing droplets required for the jump to ensue were found when varying the surface structure underneath the condensing droplets. A surface energy analysis is proposed to account for the suppression of the droplet-jumping performance in the presence of microstructures. On other hand, by impregnating the same SHSs with a low surface tension oil, i.e., SLIPSs, the adhesion between the condensate and the SLIPSs can be further reduced. On SLIPSs slight differences on the droplet density over time and shedding performance upon the inclusion of microstructures were observed. Droplets were found to shed faster and with smaller diameters on SLIPSs in the presence of microstructures when compared to solely nanostructured SLIPSs. We conclude that on SHSs the droplet-jumping performance of micrometer droplets is deteriorated in the presence of microstructures with the consequent decrease in the heat transfer performance, whereas on SLIPSs the droplet self-removal is actually improved in the presence of microstructures.