This thesis examines the role of fast/slow dynamics in understanding the mechanisms behind oscillatory patterns found in paleoclimate data. Fast/slow systems often exhibit rapid transitions between metastable states, and understanding these transitions is important to understanding climate phenomena. However, these rapid changes in the state of the system implicitly require examining trajectories that enter a region of phase space where the basic theory used to analyze fast/slow systems no longer applies. The content of this thesis examines the non-standard behavior arising from the break down of the theory, typically appearing in the form of small amplitude oscillations due to canard trajectories. First, canard theory is extended to piecewise-smooth systems. Conditions are found in which canard behavior is similar to that of smooth systems. Additionally, the dynamics are classified when these conditions are not met. Second, the new theory is used to analyze a variation on Stommel's model of large-scale ocean circulation, showing that the model is capable of exhibiting both canards and relaxation oscillations. Another variation of Stommel's model with an extra phase-space dimension is also demonstrated to exhibit relaxation oscillations. Finally, a model for glacial-interglacial cycles is analyzed through the lens of mixed-mode oscillations. The model is demonstrated to exhibit complicated oscillations due to a generalized canard phenomenon.