Nanomagnets of controlled size, organized into regular patterns open new perspectives in the fields of nanoelectronics, spintronics, and quantum computation. Self-assembling processes on various types of substrates allow designing fine-structured architectures and tuning of their magnetic properties. Here, starting from a description of fundamental magnetic interactions at the nanoscale, we review recent experimental approaches to fabricate zero-, one-, and two-dimensional magnetic particle arrays with dimensions reduced to the atomic limit and unprecedented areal density. We describe systems composed of individual magnetic atoms, metal-organic networks, metal wires, and bimetallic particles, as well as strategies to control their magnetic moment, anisotropy, and temperature-dependent magnetic behavior. The investigation of self-assembled subnanometer magnetic particles leads to significant progress in the design of fundamental and functional aspects, mutual interactions among the magnetic units, and their coupling with the environment.