Topology is one of the most recent branches of mathematics and has entered fully into one of the most modern aspects of theoretical physics: quantum computation. In this colloquium an elementary approach to the role of topology in quantum physics and its implications for quantum computing is provided. Topology helps to solve the essential problem of quantum computation: to battle its fragility in order to benefit from its enormous potential possibilities.   Topological color codes are discussed as an example of topological protection based on ground state properties. A review of their experimental implementations, including ion traps, superconducting qubits, and neutral atoms in optical lattices, is presented, along with an examination of future challenges.