The research presented in this thesis is divided into two parts. In the first part, we propose response signatures for quantum phase transitions in superconducting and bilayer graphene systems. In superconducting systems, there is the promise for realizing a Majorana quasiparticle: a fermion that is its own antiparticle and possesses some of the non-Abelian braiding statistics required to form a topological quantum computer. We propose conductance and noise signatures showing the presence of Majorana fermions in topological insulator-superconductor heterostructure Josephson Junctions. We then move on to address the possibility of realizing the physics of quantum point contacts in graphene bilayers. There have been numerous theoretical proposals of quasi-topological domain walls in bilayer graphene. Using bosonization and renormalization group considerations, we propose transport signatures characterizing the pinch-off behavior of the effective point contact formed by the intersection of two domain walls. In the second part of this thesis, we provide several examples of nodal band features protected by the combination of topology and crystalline geometry. Nodal features in condensed matter systems manifest when the Fermi surface consists only of a limited set of band-touching points with fixed dispersion. The low-energy theories of these touching points resemble those in particle physics, and these nodes, such as the quintessential examples in graphene, are therefore frequently characterized with names such as Dirac and Weyl fermions. The presence of nodal band features at the Fermi energy can have unique implications for bulk transport and surface physics, and so there has been a great effort in recent years to find new theoretical and real-material examples of nodal systems. We begin by showing that when spin-orbit interaction is weak, the same Z_2 invariant that predicts a topological insulator can be used when inversion symmetry is present to predict topological Dirac line nodes in crystal systems. On the surface of these line node semimetals, the projected interior of the line nodes can host a two-dimensional nearly-flat band, and could provide a route towards experimental access of phases with significant electron-electron interactions. We then present the first known example of a nodal condensed matter system with a description beyond particle physics: the double Dirac semimetal. In these systems, eightfold-degenerate linearly-dispersing nodal points manifest at the Brillouin zone edge. We list all possible space groups that can host double Dirac points when spin-orbit interaction is non-negligible, and we show that the expanded set of time-reversal-symmetric mass terms for these new fermions allows for new routes towards strain-engineering topological phase transitions and also provides the possibility of topologically-nontrivial line defects. We then move on to two dimensions, for which we use a consideration of compact flat manifolds to deduce all possible manifestations of nodal physics in strong spin-orbit systems. Through this analysis, we explain in more general terms some of the more exotic examples of nodal systems proposed over the past few years, and predict new examples in two and three dimensions. Using conclusions from this analysis and specializing to the wallpaper groups, we then show that a consideration of minimal insulating filling allows one to exhaustively characterize all possible topological and topological crystalline insulators. By realizing that the limited set of wallpaper groups constrains the Wilson-loop eigenvalue flows of a three-dimensional bulk insulating crystal, we present the discovery of a new topological crystalline insulator: the topological Dirac insulator. Unlike the surface states of a conventional topological insulator, the surface states of this new insulator are fourfold degenerate, and therefore can be gapped to realize truly topological surface quantum spin Hall domain walls. Finally, we present the first example of a filling-enforced semimetal in a magnetic system. By exploiting the modified time-reversal symmetry in certain antiferromagnetic systems, we characterize a new class of two-dimensional magnetic Dirac semimetals. We show that these semimetals manifest a new quantum critical point between quantum Hall phases, and discuss their place in the context of fermion doubling theorems.