  In this Perspective article we report on recent progress on studies of
synthetic dimensions, mostly, but not only, based on the research realized
around the Barcelona groups (ICFO, UAB), Donostia (DIPC), Pozna\'n (UAM),
Krak\'ow (UJ), and Allahabad (HRI). The concept of synthetic dimensions works
particularly well in atomic physics, quantum optics, and photonics, where the
internal degrees of freedom (Zeeman sublevels of the ground state, metastable
excited states, or motional states for atoms, and angular momentum states or
transverse modes for photons) provide the synthetic space. We describe our
attempts to design quantum simulators with synthetic dimensions, to mimic
curved spaces, artificial gauge fields, lattice gauge theories, twistronics,
quantum random walks, and more.
