The precise measurement of cosmic-ray antinuclei serves as an important means for identifying the nature of dark matter and other new astrophysical phenomena, and could be used with other cosmic-ray species to understand cosmic-ray production and propagation in the Galaxy. For instance, low-energy antideuterons would provide a "smoking gun"signature of dark matter annihilation or decay, essentially free of astrophysical background. Studies in recent years have emphasized that models for cosmic-ray antideuterons must be considered together with the abundant cosmic antiprotons and any potential observation of antihelium. Therefore, a second dedicated Antideuteron Workshop was organized at UCLA in March 2019, bringing together a community of theorists and experimentalists to review the status of current observations of cosmic-ray antinuclei, the theoretical work towards understanding these signatures, and the potential of upcoming measurements to illuminate ongoing controversies. This review aims to synthesize this recent work and present implications for the upcoming decade of antinuclei observations and searches. This includes discussion of a possible dark matter signature in the AMS-02 antiproton spectrum, the most recent limits from BESS Polar-II on the cosmic antideuteron flux, and reports of candidate antihelium events by AMS-02; recent collider and cosmic-ray measurements relevant for antinuclei production models; the state of cosmic-ray transport models in light of AMS-02 and Voyager data; and the prospects for upcoming experiments, such as GAPS. This provides a roadmap for progress on cosmic antinuclei signatures of dark matter in the coming years.