Electronic systems subject to competing interactions can end up in different phases as the balance between these interactions shifts. When a quantum critical point separates these phases, exotic electronic behavior often marks the vicinity of the transition. In this work, we construct nanopatterned devices to probe such critical phenomena. The basic element of our devices is the GaAs/AlGaAs quantum dot, an isolated region of electronic charge which is coupled to a two-dimensional gas of weakly interacting electrons. We use different designs of quantum dots to realize different models. The first device studied in this work realizes the spin two-channel Kondo ('spin 2CK') model. In this model, a single impurity (i.e. a single spin-degenerate dot) is coupled to two separate reservoirs. When the couplings to both reservoirs are unequal, the more strongly coupled reservoir screens the impurity spin degeneracy and forms a many-body singlet; this is known as the Kondo effect. When both reservoirs are coupled equally strongly, a non-Fermi liquid ground state arises as a result of the overscreening by both reservoirs. We probe the anomalous scaling properties of this state, and show how it transitions into a more conventional Fermi liquid under the influence of various perturbations. The second device is first operated as a single metallic quantum dot in the quantum Hall regime. Spin degeneracy is broken, but the device can be tuned such that there is now a charge degeneracy which can then be screened by coupling to a reservoir. We tune to and away from equal couplings to see the effect of the two-channel Kondo state. Finally, we operate the second device in its full form as a double-dot device, to explore the competition between dot-lead and interdot interactions.