Quantum computers promise the numerically exact simulation of molecules and materials. Furthermore, they are amongst the algorithms that have the lowest resource requirements for surpassing the power of classical computers. In this talk, I will briefly introduce the basic concepts of quantum computing and quantum simulation. Then, I will review the recent rapid progress in developing more efficient algorithms that have been achieved by many researchers in the field including our research group. I will describe the families of available algorithms (phase estimation, adiabatic and variational quantum eigensolver approaches) as well as the status of several experimental implementations of them either carried out or underway. These implementations span most of the currently available quantum architectures including quantum optics, ion traps, NV centers and superconducting quantum bits. I will provide a prelude of the relevance of these applications to society and will conclude with the prospects of the field.