Cats (Felis catus) are among the most damaging invasive predators in the world, and their impacts in Aotearoa New Zealand (NZ) are particularly severe. However, unlike the invasive predators that are targeted for eradication under the Predator Free NZ initiative, cats are also highly valued by people and therefore will likely remain widespread in NZ for the foreseeable future. This raises the question of how to manage the impacts of cats, which include predation, competition, and disease affecting native species, livestock, and humans. Appropriate management actions will depend on land use (e.g. urban areas vs wilderness), the values to be protected (e.g. wildlife, human health), as well as safety, humaneness, social acceptability, and cost-effectiveness. We review current knowledge on the impacts and management of cats in NZ and overseas, identify knowledge gaps preventing effective management, and suggest approaches for research to address these gaps. Our suggested research priorities include: (1) improved methods for monitoring cats and their impacts on natural, social and economic values, (2) development of humane, effective, and socially acceptable methods to manage the impacts of cats, (3) engagement with cat owners to improve outcomes for cats, people, and the environment, and (4) investigating potential indirect ecological effects of cat control, such as ecological release of prey or competitors.