Thermal conductivity and heat capacity have been established as powerful bulk probes for studying superconducting states which go beyond the standard BCS theory of superconductivity. In this thesis, I present transport and thermodynamic measurements in ultra-low temperatures (down to 20~mK) on two different unconventional superconducting systems: AFe$_2$As$_2$ (A=K,Rb,Cs) and UTe$_2$.

In the first study, I present measurements of charge and thermal transport in the iron-based superconductor KFe$_2$As$_2$ in magnetic fields aligned precisely in the $ab-$plane. This allows the study thermal transport in a Pauli-limited superconductor with the quasiparticle mean free path controlled with disorder induced by electron irradiation. In KFe$_2$As$_2$ with a high level of disorder, we observe a residual quasiparticle thermal conductivity which varies linearly in field, which is consistent with predictions for a paramagnetically limited superconductor. None of the measured samples display signatures of a first-order superconducting transition or the Fulde-Ferrell-Larkin-Ovchinikov (FFLO) phase.

In the field-induced normal state in a pristine sample of KFe$_2$As$2$, we found non-Fermi liquid temperature dependence of the charge and thermal resistivity. Furthermore, the thermal resistivity indicates the persistence of strong inelastic quasiparticle scattering in the superconducting state for fields below $H{c2}$. The resistivity was measured in fields up to 15~T along both the $a$ and $b$ axes in the pristine and irradited KFe$_2$As$_2$ samples, as well as in RbFe$_2$As$_2$ and CsFe$_2$As$_2$. No indication of field tuning of the non-Fermi liquid behavior was observed. Instead, the field dependent resistivity in these samples follows a generalized Kohler scaling, indicating that the scattering mechanism giving rise to non-Femi liquid resistivity in zero field is unaffected by the application of magnetic field.

Finally, I present measurements of thermal transport and heat capacity in UTe$_2$ to resolve the superconducting gap structure. Surprisingly, no residual quasiparticle thermal conductivity is observed despite a large residual density of states observed in heat capacity. The rapid increase of residual quasiparticle thermal conductivity in magnetic field, as well as the low temperature power law dependence of both heat capacity and magnetic penetration depth, suggest a point nodal gap structure in UTe$_2$. Furthermore, careful measurements of the heat capacity in the vicinity of the superconducting transition uncover a splitting of the superconducting transition temperature indicative of two nearly degenerate superconducting phases present in UTe$_2$.