Cells compartmentalize parts of their interiors into liquid-like condensates, which can be reconstituted in vitro. Although these condensates interact with membrane-bound organelles, their potential for membrane remodeling and the underlying mechanisms of such interactions are not well-understood. Here, we demonstrate that interactions between protein condensates - including hollow ones, and membranes can lead to remarkable morphological transformations and provide a theoretical framework to describe them. Modulation of solution salinity or membrane composition drives the condensate-membrane system through two wetting transitions, from dewetting, through a broad regime of partial wetting, to complete wetting. When sufficient membrane area is available, fingering or ruffling of the condensate-membrane interface is observed, an intriguing phenomenon producing intricately curved structures. The observed morphologies are governed by the interplay of adhesion, membrane elasticity, and interfacial tension. Our results highlight the relevance of wetting in cell biology, and pave the way for the design of synthetic membrane-droplet based biomaterials and compartments with tunable properties.