**Universal quantum simulators I**

The first significant impact of quantum computing will be universal quantum simulation, which will compute otherwise hard-to-evaluate properties of Hamiltonian and open quantum systems. Quantum simulators that exceed classical computing capability only require dozens of qubits and dozens of gates so could be feasible in a few years if quantum error correction can be neglected. In these lectures we introduce the concept and theory of the universal quantum simulator and recent experimental advances with "analogue" and "digital" quantum simulators. In addition we will see how the universal quantum simulator can simulate the evolution of a state, explore ground-state properties and be used to solve giant systems of coupled linear equations.