Symmetry and topology are fundamental properties of nature. Mathematics provides us with a general framework to understand these concepts. On one side, symmetry describes the invariance properties of an object for specific transformations. On the other side, topology classifies objects under continuous deformations. Two objects with different topologies cannot be deformed one into each other without creating or annihilating a singularity, sometimes referred to as 'node'.

These concepts gradually found applications in physics, namely in the description of the electronic properties of solids, which is the focus of this thesis. Symmetry and topology protect special nodes in the band structure of a crystal, where several states are degenerate in energy. Close to a node, the electron wave function obeys the Dirac or Weyl Hamiltonians, which were formally introduced to describe fermions in high-energy particle physics. These exotic fermions exhibit unique optical and transport properties, fully manifesting their quantum nature. Symmetry guides us in the search for which classes of materials may host topological nodes. Recently the attention of the scientific community has been attracted by crystals with non-symmorphic symmetries (screw axes and glide planes), as promising candidates for topological phases.

This thesis focuses on two families of non-symmorphic crystals, whose topological properties have been investigated by combining conventional angle-resolved photoelectron spectroscopy (ARPES) with state-of-the-art spin-resolved (spARPES) and time-resolved ARPES (trARPES). The experimental results are supported by calculations carried out in collaboration with theory groups.

ZrSiTe and ZrSiSe belong to the same class of non-trivial semimetals. They host Dirac electrons with large mobility. The results of my spARPES experiments clarify that spin-orbit interaction (SOI) not only removes the topological nodes in ZrSiTe, but also induces a 'hidden' spin polarization of its bulk electronic states, otherwise forbidden by the inversion symmetry of the lattice. Moreover, trARPES data provide evidence that the electron-electron interaction is only partially screened in the metallic state of ZrSiSe. As a consequence, the band velocity is enhanced, at odds with general expectations. More importantly, I show that this band renormalization can be controlled at the ultrafast time scale (namely fs scale, with 1 fs=10^{-15} s) via intense optical excitation, paving the way for engineering the band structure of Dirac semimetals.

Tellurium is a chiral semiconductor, with a small and direct band gap. The low-symmetry of its lattice and the simple chemical composition make it the ideal case to study the interplay between symmetry and topology. With ARPES I determine several Weyl nodes in its electronic structure. By means of spARPES, I demonstrate that in their surrounding the spin exhibits a hedgehog configuration. This observation is new and it highlights the connection between spin-dependent and topology-related properties in Te. Finally, I illustrate promising preliminary results based on trARPES that explore the appealing possibility of optically controlling the topology of the electronic structure of Te upon excitation of coherent phonons.