We demonstrate four-and two-terminal perovskite-perovskite tandem solar cells with ideally matched band gaps. We develop an infrared-absorbing 1.2-electron volt band-gap perovskite, FA(0.75)Cs(0.25)Sn(0.5)Pb(0.5)I(3), that can deliver 14.8% efficiency. By combining this material with a wider-band gap FA(0.83)Cs(0.17)Pb(I0.5Br0.5)(3) material, we achieve monolithic two-terminal tandem efficiencies of 17.0% with > 1.65-volt open-circuit voltage. We also make mechanically stacked four-terminal tandem cells and obtain 20.3% efficiency. Notably, we find that our infrared-absorbing perovskite cells exhibit excellent thermal and atmospheric stability, not previously achieved for Sn-based perovskites. This device architecture and materials set will enable "all-perovskite" thin-film solar cells to reach the highest efficiencies in the long term at the lowest costs. ; We thank M. T. Horantner for performing the Shockley-Queisser calculation. The research leading to these results has received funding from the Graphene Flagship (Horizon 2020 grant no. 696656 - GrapheneCore1), the Leverhulme Trust (grant RL-2012-001), the UK Engineering and Physical Sciences Research Council (grant EP/J009857/1 and EP/M020517/1), and the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement nos. 239578 (ALIGN) and 604032 (MESO). T.L. is funded by a Marie Sklodowska Curie International Fellowship under grant agreement H2O2IF-GA-2015-659225. A.B. is financed by IMEC (Leuven) in the framework of a joint Ph.D. program with Hasselt University. B.C. is a postdoctoral research fellow of the Research Fund Flanders (FWO). We also acknowledge the U.S. Office of Naval Research for support. We acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility (http://dx.doi.org/10.5281/zenodo.22558) and the ARCHER UK National Super-computing Service under the "AMSEC" Leadership project. We thank the Global Climate and Energy Project (GCEP) at Stanford University. All data pertaining to the conclusions of this work can be found in the main paper and the supplementary materials.