In this thesis, I describe research results in three topics: a test for the chiral anomaly, the anomalous Nernst effect (ANE) and the anomalous Hall effect (AHE) in Weyl and Dirac semimetals, and a novel zero-Hall state observed in the quantum limit of non-symmorphic material KHgSb.

Experimentally, direct evidence for the chiral anomaly has already been observed in the longitudinal magnetoresistance in Na_{3}Bi and GdPtBi. However, the possibility of current jetting has raised general concerns about LMR results. Here, we design and perform a litmus test, the squeeze test, that allows the intrinsic LMR in Na_{3}Bi and GdPtBi to be sharply distinguished from pure current jetting effects using pure Bi as a counter example. A second litmus test we provide is the parametric plot of the planar angular magnetoresistance. By plotting the diagonal resistivity component vs. the off-diagonal resistivity component we can easily distinguish between two cases. Our litmus tests can work as standard tests that can distinguish the LMR intrinsic to the chiral anomaly from current jetting.

The Berry curvature is another important property of Weyl nodes. We observed strong AHE in the Dirac semimetal Na_{3}Bi and ANE in the Weyl semimetal NbAs. The step-like functions of AHE and ANE originate from the anomalous velocity which results from the Berry curvature at the Weyl nodes. Our observation of AHE and ANE uncovers the geometrical property of Weyl nodes in momentum space.

Another attractive field of research in topological matter is the protection of exotic states by symmetry. The large class of non-symmorphic materials has recently been predicted to hold unusual surface states. KHgSb is one example of them. It is predicted to have double quantum spin Hall states. Here we will discuss our observation of gap-protected zero-Hall states in KHgSb in a strong magnetic field *B*. We propose that, in the quantum limit, the chemical potential drops into the bulk gap, intersecting equal numbers of right- and left-moving quantum spin Hall surface modes to produce the zero-Hall state. This zero-Hall state is a direct result of the glide symmetry in a non-symmorphic material.