In these two lectures, I will introduce the main algorithms used in today’s noisy and tomorrow’s fault-tolerant quantum computers. After a quick introduction to gate-based quantum computation, I will review basic primitives like the quantum Fourier transform and their use in algorithms such as quantum phase estimation, with applications to the factoring problem (Shor’s algorithm) and energy estimation in quantum physics. Then, I will turn to the challenges of decoherence in quantum computers, to the variational algorithms that have been designed to mitigate its effects (including the variational quantum eigensolver, VQE), and to their limitations and some counter-measures like error mitigation. In the hands-on session, we will implement a phase estimation algorithm as well as a VQE algorithm applied to a quantum chemistry problem.