We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-like exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes as limiting cases the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the inter-impurity tunneling-coupling. We point out that due to the existence of exponentially low temperature scales, there is a parameter range where the stable "zero-temperature" fixed point is essentially never reached (not even in numerical renormalization group calculations). The "zero-temperature" conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.