Deforestation and other land-use changes have released ~150 billion metric tonnes of carbon to the atmosphere since 1850. Reversing forest losses through restoration, improvement, and conservation is a critical policy goal for greenhouse-gas (GHG) mitigation. Here, we examine three questions about forest mitigation, including some ecological, demographic, and economic constraints on those opportunities: 1) Where will forest lands come from to help slow the buildup of greenhouse gases? 2) To what extent can future management, including genetic engineering, extend productivity beyond what native and managed forests already provide? and 3) What will be the co-effects of maximizing forest productivity for water, nutrients, biodiversity, and other ecosystem attributes? Efforts to reduce deforestation and degradation (REDD) could cut global deforestation rates in half by 2030, preserving 1.5 to 3 billion metric tons of CO2-equivalent (tCO2e) emissions per year depending on the economic incentives. One possible constraint on this and other forest activities is landowner behavior, including competition with agriculture, urban development, and other land uses. Our new economic modeling for the United States suggests that offset payments there could reduce GHG emissions by as much as 900 million tCO2e per year through afforestation, forest management practices, and bioelectricity generation. However, proposed carbon payments also reduce agricultural land area in the U.S. by a further 10% or more, decreasing agricultural exports and increasing commodity food prices, imports, and the potential for leakage - driving land use change in other regions. In addition to more land area needed for forest mitigation, more forest productivity per area is also needed. Selective breeding and genetic engineering can improve productivity per area, resistance to insects and pathogens, and product quality. Maximizing forest productivity also carries some potential risks, however, including invasions, freshwater and biodiversity losses, fires, and transgene spread. We illustrate these opportunities and risks using the projected commercialization of transgenic eucalypts in the United States. Key goals of forest mitigation activities are to preserve and restore forests and to improve forest management for the benefit of climate, while maintaining the production of wood, fiber, and other goods. At the same time we need to preserve, grow, and manage forests for biodiversity and ecosystem services in general. Striking that balance will help us realize the benefits of forestry for climate mitigation, for people, and for other species on Earth.