Gauge theories are ubiquitous in physics.Many intriguing phenomena in condensed matter physics owe to the action of the electromagnetic field, which is an Abelian gauge theory.The numerical treatment of many-body systems is inherently complex due to the exponentially growing size of the Hilbert space.While in one dimension an area law guarantees that numerical methods on classical computers can deal with strongly correlated systems, in higher dimensions the quantum simulation comes as the panacea for the many-body problem.The present thesis comprises the elaboration of experimentally feasible methods for the quantum simulation of dynamical Abelian gauge fields with ultra-cold gases of neutral atoms and the theoretical analysis of the related model Hamiltonians.As neutral atoms do not interact with external vector potentials like charged particles would do, the gauge fields have to be artificially engineered.The elements of a gauge theory that need to be replicated on a quantum simulator vary depending on the subject of investigation.The key ingredient at the root of many condensed matter phenomena, from the quantum Hall effect to superconductivity and chiral topological insulators, is the Berry phase.Whilst artificial static gauge fields have been widely explored, much remains to do regarding the realization of artificial dynamical gauge fields.In Chapter 3 we present a method based on the amplitude modulation of a one-dimensional optical lattice, which allows for an unprecedented degree of control over a wide range of parameters.The method also comprises the generation of a density-dependent complex phase, fundamental to the creation of anyonic pseudo-particles.The anyons are amenable of observation through interferometric measurement, realizable with the same experimental set-up.With regard to gauge theories, the Berry phase is just the visible tip of the iceberg.Below the waterline, there is more to consider in order to comprehensively reproduce a gauge theory, like the electric and magnetic fields in quantum electrodynamics.Moreover, a full account for the inherent symmetry is crucial to investigate phenomena proper of non-Abelian gauge theories in the context of high-energy physics, such as confinement.For this collection of topics, one can turn to lattice gauge theories.In Chapter 5, we consider a class of lattice gauge theories particularly suitable for quantum simulation, the Quantum Link Model.The study of the Abelian U(1) Quantum Link Model on a ladder geometry reveals a highly non-trivial phase diagram, featuring a symmetry-protected topological phase.In both Chapters, innovative solutions for the experimental realization of the model Hamiltonians are designed and proposed.To gain numerical access to the ground-state properties and the dynamics of the systems investigated we make use of state-of-the-art numerical methods based on Tensor Networks.The elements of the numerical analysis carried out throughout this thesis are presented in Chapter 6.In the last part we offer an outlook on research perspectives related to the topics discussed in the thesis.