A 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.