Many biomolecular condensates are liquid-like droplets composed of proteins and/or RNAs. A key mechanical property for any liquid droplet is its surface tension. Here, I will discuss two sets of surface tension phenomena involving biomolecular condensates. First, there is now increasing evidence that biomolecular condensates interact with other cellular components, such as lipid membranes, for important biological functions. For example, during seed development of the plant Arabidopsis thaliana, micrometer-sized condensates form within the vacuolar lumen and wet the tonoplast. Distinct tonoplast shapes arise in response to membrane wetting by condensates, including membrane budding and membrane nanotubes. Another example is the formation of capillary bridges between lipid vesicles by protein condensates, as recently observed e.g. in damaged mitochondria of HeLa cells. Studying the morphology of the capillary bridges, we find three distinct morphologies (bridging, enclosing and zipping), each with distinct mechanical responses. Second, given the relevance of surface tension for wide-ranging cellular processes, there is a need to measure its value in living cells. Correspondingly, we have recently developed a high-throughput flicker spectroscopy approach to calculate the surface tension of thousands of condensates. Demonstrating this approach on stress granules, we discovered that a surface tension-only model is inadequate for describing stress granules in live cells. We find that the measured fluctuation spectra require an additional bending rigidity parameter, which supports the view that stress granules are viscoelastic droplets with a structured interface. Moreover, we observe that the measured interfacial tensions and bending rigidities span a range of several orders of magnitude. Hence, different types of stress granules (and more generally, other biomolecular condensates) can only be differentiated via large-scale surveys.