In our laboratories, we produce the coldest matter of the universe at temperatures just a few billionths of a degree above absolute zero. Under such unimaginably cold conditions, the elusive quantum world, typically masked at higher temperatures, comes into focus. Ultracold matter offers unique opportunities for studying quantum many-body problems that are relevant to fields as diverse as condensed matter physics, statistical physics, quantum chemistry, and high-energy physics. In particular, the unprecedented degree of controllability and the availability of advanced tools, which allow the manipulation and the detection at the level of single atoms, make these ultracold systems ideal as analog quantum simulators. They realize the Feynman’s conjecture: “Nature isn’t classical . . . and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy”. Our group is a world leader in the study of ultracold quantum matter composed by fermionic and bosonic atoms. We investigate quantum atomic mixtures and very recently we have at our disposal a hybrid atom-ion mixture, in which we marry two usually well-separated research fields. We manipulate these systems with high spatial resolution in optical arbitrarily tailored potentials, lattices, even in low-dimensional geometries. In collaboration with our theory group, we investigate fundamental quantum effects, quantum interferometry, quantum transport phenomena in the presence of strong correlations, superfluid phases in fermionic mixtures, up to the propagation of chiral currents driven by synthetic gauge fields. At the same time, we disclose and characterize new states of matter, such as quantum droplets and the long sought supersolid phase.