Fault-tolerant quantum computation with high threshold in two dimensions Quantum computation is fragile. Exotic quantum states are created in the process, exhibiting entanglement among large number of particles across macroscopic distances. In realistic physical systems, decoherence acts to transform these states into more classical ones, compromising their computational power. However, decoherence can be counteracted by quantum error-correction. In my talk I first give an introduction to fault-tolerant quantum computation in the setting of two-dimensional lattices of qubits in which only nearest neighbors may interact. Such a geometric constraint is, in many physical systems considered for building a large-scale quantum computer, imposed by experimental reality. It is relevant for arrays of superconducting qubits, optical lattices and also for segmented ion traps. Efficient solutions for achieving fault-tolerance in such a scenario are topological. I will review some of the known constructions based on surface codes and color codes.