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During endocytosis of nanoparticles by cells, the cellular membranes engulf the particles, thereby forming a closed membrane neck that subsequently undergoes fission. For solid nanoparticles, these endocytic processes have been studied in some detail. Recently, such processes have also been found for liquid and condensate droplets, both in vitro and in vivo. These processes start with the spreading of the droplet onto the membrane followed by partial or complete engulfment of the droplet. Here, we use molecular dynamics simulations to study these processes at the nanoscale, for nano-sized droplets and vesicles. For both partial and complete engulfment, we observe two different endocytic pathways. Complete engulfment leads to a closed membrane neck which may be formed in a circular or strongly non-circular manner. A closed circular neck undergoes fission, thereby generating two nested daughter …