Molecular nanomagnets, often called single-molecule magnets (SMMs), have attracted much interest in recent years both from the experimental and theoretical points of view. These systems are organometallic clusters characterized by a large spin ground state with a predominant uniaxial anisotropy. The quantum nature of these systems makes them very appealing for phenomena occurring on the mesoscopic scale, i.e. at the boundary between classical and quantum physics. Below their blocking temperature, they exhibit magnetization hysteresis, the classical macroscale property of a magnet, as well as quantum tunneling of magnetization (QTM) and quantum phase interference, the properties of a microscale entity. QTM is advantageous for some potential applications of SMMs, e.g. in providing the quantum superposition of states for quantum computing, but is a disadvantage in others such as information storage. It is widely admitted that SMMs have a potential for quantum computation, in particular because they are extremely small and almost identical, allowing one to obtain, in a single measurement, statistical averages of a larger number of qubits. This review introduces few basic concepts that are needed to understand the quantum phenomena observed in molecular nanomagnets and it concludes by mentioning new trends of the field of molecular nanomagnets towards molecular spintronics.