  The past few years have witnessed the concrete and fast spreading of quantum
technologies for practical computation and simulation. In particular, quantum
computing platforms based on either trapped ions or superconducting qubits have
become available for simulations and benchmarking, with up to few tens of
qubits that can be reliably initialized, controlled, and measured. The present
review aims at giving a comprehensive outlook on the state of art capabilities
offered from these near-term noisy devices as universal quantum simulators,
i.e. programmable quantum computers potentially able to calculate the time
evolution of many physical models. First, we give a pedagogic overview on the
basic theoretical background pertaining digital quantum simulations, with a
focus on hardware-dependent mapping of spin-type Hamiltonians into the
corresponding quantum circuit model as a key initial step towards simulating
more complex models. Then, we review the main experimental achievements
obtained in the last decade regarding the digital quantum simulation of such
spin models, mostly employing the two leading quantum architectures. We compare
their performances and outline future challenges, also in view of prospective
hybrid technologies, towards the ultimate goal of reaching the long sought
quantum advantage for the simulation of complex many body models in the
physical sciences.
