Delivering the Sustainable Development Goals (SDGs) requires balancing demands on land between agriculture (SDG 2) and biodiversity (SDG 15). The production of vegetable oils and, in particular, palm oil, illustrates these competing demands and trade-offs. Palm oil accounts for ~40% of the current global annual demand for vegetable oil as food, animal feed and fuel (210 Mt), but planted oil palm covers less than 5-5.5% of the total global oil crop area (approximately 425 Mha) due to oil palm's relatively high yields. Recent oil palm expansion in forested regions of Borneo, Sumatra and the Malay Peninsula, where >90% of global palm oil is produced, has led to substantial concern around oil palm's role in deforestation. Oil palm expansion's direct contribution to regional tropical deforestation varies widely, ranging from an estimated 3% in West Africa to 50% in Malaysian Borneo. Oil palm is also implicated in peatland draining and burning in Southeast Asia. Documented negative environmental impacts from such expansion include biodiversity declines, greenhouse gas emissions and air pollution. However, oil palm generally produces more oil per area than other oil crops, is often economically viable in sites unsuitable for most other crops and generates considerable wealth for at least some actors. Global demand for vegetable oils is projected to increase by 46% by 2050. Meeting this demand through additional expansion of oil palm versus other vegetable oil crops will lead to substantial differential effects on biodiversity, food security, climate change, land degradation and livelihoods. Our Review highlights that although substantial gaps remain in our understanding of the relationship between the environmental, socio-cultural and economic impacts of oil palm, and the scope, stringency and effectiveness of initiatives to address these, there has been little research into the impacts and trade-offs of other vegetable oil crops. Greater research attention needs to be given to investigating the impacts of palm oil production compared to alternatives for the trade-offs to be assessed at a global scale.