This talk is concerned with Triebel-Lizorkin Spaces, which are advanced generalizations of Sobolev Spaces. We will see that this function spaces can be used to measure the smoothness and integrability properties of functions and distributions in a very precise way. Since Triebel- Lizorkin Spaces can be described via several equivalent quasi-norms using the Fourier transform, higher order differences, oscillations, wavelets, atoms or quarklets, they are a powerful tool when solving problems stemming from very different areas of analysis. For example they can be used within regularity theory for PDEs to describe the smoothness and integrability properties of unknown solutions. Here it is also possible to investigate demanding PDEs such as the Schrödinger equation or to deal with SPDEs. Using equivalent quarklet characterizations Triebel-Lizorkin Spaces also can be applied to approximate the unknown solution of PDEs by adaptive numerical methods inspired by finite element methods. Finally we will discuss how Triebel-Lizorkin Spaces can be used within the theory of dynamical systems and in connection with neural networks.