Trimble, Christie JordanA Josephson junction (JJ) couples the supercurrent flowing between two weakly linked superconductors to the phase difference between them via a tunnel barrier, giving rise to a current-phase relation (CPR). While a sinusoidal CPR is expected for conventional junctions with insulating weak links, devices made from some exotic materials may give rise to unconventional CPRs and unusual Josephson effects. Here, I experimentally investigate three such cases. In the first part of the thesis, I fabricate JJs with weak links made of the topological crystalline insulator Pb$_{0.5}$Sn$_{0.5}$Te and compare them with JJs made from its topologically trivial cousin, PbTe. I find that measurements of the AC Josephson effect reveal a stark difference between the two: while the PbTe JJs exhibit Shapiro steps at the expected values of $V=nhf/2e$, Pb$_{0.5}$Sn$_{0.5}$Te JJs show more complicated subharmonic structure. I present the skewed sinusoidal CPR necessary to reproduce these measurements and discuss a potential origin for this alteration. Next, I investigate the proximity-induced superconductivity in SnTe nanowires by incorporating them as weak links in Josephson junctions. I report indications of an unexpected breaking of time-reversal symmetry in these devices, including observations of an asymmetric critical current in the DC Josephson effect, a prominent second harmonic in the AC Josephson effect, and a magnetic diffraction pattern with a minimum in critical current at zero magnetic field. I analyze how multiband effects and the experimentally visualized ferroelectric domain walls may give rise to a nonstandard CPR in the junction. Finally, I measure JJs with weak links made of the topological insulator (BiSb)$_2$Te$_3$. Under low frequency RF radiation, I observe suppression of the first and third Shapiro steps, consistent with the fractional AC Josephson effect. This could indicate a 4$\pi$ periodic component in the junction's CPR, potentially implying the presence of Majorana bound states. However, not all of the devices showed this behavior; some devices show suppression of only the first step, while others show distortions to the AC Josephson effect which differ upon repeated measurements, possibly indicating other nonequilibrium effects at play. I discuss this behavior and possible topologically trivial sources of step suppression found in the literature.enOn the nature of the Josephson effect in topologically nontrivial materialsDissertationCondensed matter physicsJosephson junctionMultiband superconductivityShapiro stepsSuperconductivityTopological insulators