Complex architectures of nanostructures are routinely elaborated using bottom-up or nanofabrication processes. This technological capability allows scientists to engineer materials with properties that do not exist in nature, but also to manufacture model systems to explore fundamental issues in condensed matter physics. Two-dimensional frustrated arrays of magnetic nanostructures are one class of systems for which theoretical predictions can be tested experimentally. These systems have been the subject of intense research in the last few years and allowed the investigation of a rich physics and fascinating phenomena, such as the exploration of the extensively degenerate ground-state manifolds of spin ice systems, the evidence of new magnetic phases in purely two-dimensional lattices, and the observation of pseudoexcitations involving classical analogues of magnetic monopoles. We show here, experimentally and theoretically, that simple magnetic geometries can lead to unconventional, non-collinear spin textures. For example, kagome arrays of inplane magnetized nano-islands do not show magnetic order. Instead, these systems are characterized by spin textures with intriguing properties, such as chirality, coexistence of magnetic order and disorder, and charge crystallization. Magnetic frustration effects in lithographically patterned kagome arrays of nanomagnets with out-of-plane magnetization also lead to an unusal, and still unknown, magnetic ground state manifold. Besides the influence of the lattice geometry, the micromagnetic nature of the elements constituting the arrays introduce the concept of chiral magnetic monopoles, bringing additional complexity into the physics of artificial frustrated spin systems.