In this colloquium I will review the theoretical research on repulsively-interacting ions confined in Paul or Penning traps and radiatively cooled to crystallization. The ions interact via the unscreened Coulomb interaction, the emerging structures can be manipulated using lasers. I will discuss these crystalline structures, highlighting analogies and difference with typical bulk structures in condensed matter physics, and discuss their properties in the quantum regime. The focus of this talk will be majorly set on the results of my collaboration with Prof. Shmuel Fishman. This colloquium is dedicated to his memory.