The interplay of interactions and quenched disorder in two-dimensional Fermi liquids has been actively studied for decades. In contrast, the role of impurities in incoherent electron systems without well-defined Landau quasiparticles, often called non-Fermi liquids, remains largely unexplored. I will present our recent work on unconventional metallic properties and superconductivity in a controlled model exhibiting non-Fermi liquid behavior near a two-dimensional quantum critical point in the presence of disorder. We show that the normal state of this system exhibits various intriguing properties, such as anomalous diffusion characterized by the dynamical scaling exponent z=4, divergent tunneling density of states, and unconventional low-temperature behavior of the conductivity and specific heat. In addition, we find that non-magnetic impurities strongly affect the interplay between pairing instability and fermionic incoherence, leading to an intermediate range of disorder strength where superconductivity is enhanced, eventually followed by a tendency towards the superconductor-insulator transition at even stronger disorder.
