Topology, an important topic in physics since several years, is handled as possible solution to many current-state problems in electronics and energy. It could allow to dramatically shrink computational devices or increase their speed without the current problem of heat dissipation, or topological principles can be used to introduce room temperature high-conduction paths within materials. Unfortunately, while many promising materials have been presented yet, the one breakthrough material is still missing. Current style materials are either consisting of toxical elements, obstructing possible use cases, or their electronic structure is too complex to investigate the interplay of all the facets of the electronic structure present in the mateirals. In this thesis, two very promising materials will be thoroughly introduced, namely TaIrTe4 and GaGeTe. Both materials have the potential, to lift one of the shortcomings mentioned. First, TaIrTe4 will be presented. TaIrTe4 is a simplistic Weyl semimetal in terms of its electronic and topological structure - the simplest yet known material. It hosts four Weyl points, the minimum amount of Weyl nodes possible in a non-centrosymmetric material. Predictions state, that these nodes are well separated throughout the Brillouin zone, and are connected by nearly parallel Fermi arcs. The existance of the topological states is proved in this thesis through angle-resolved photoemission spectroscopy (ARPES) and confirmed by spin polarization measurements on these states. GaGeTe is predicted to be a Bi2Se3-style topological insulator, but ARPES data presented shows, that no direct band gap could be observed. Yet, a topological state is still believed to be present. This makes this material interesting in many ways: its elemental composition is less toxic than bismuth and selenium, as well as it is the first realization of such a specific electronic structure. A full discussion of the electronic states close to the Fermi level including the possible existance of topological states is shown in this thesis.