The discovery of the fractional quantum Hall effect for two-dimensional electron gases immersed in a strong orthogonal magnetic field represents a cornerstone of modern physics. The states responsible for the appearance of the fractional quantum Hall effect have been found to be part of a whole new class of phases of matter, characterized by an internal order with unprecedented properties and known as topological order. This fact opened up a completely new territory for physical studies, paving the way towards many of the current hot topics in physics, such as topological phases of matter, topological order and topological quantum computing. As it happens for most topologically-ordered phases, fractional quantum Hall states are breeding ground for the observation of many exotic physical phenomena. Important examples include the appearance of degenerate ground states when the system in placed on a space with non-trivial topology, the existence of chiral gapless edge excitations which unidirectionally propagate without suffering of back-scattering processes, and the possibility of hosting elementary excitations, known as quasiparticles and quasiholes, carrying fractional charge and anyonic statistics. Even though for years since their discovery fractional quantum Hall states have been studied only in electronic systems, the recent advances made in the domains of quantum simulators and artificial gauge fields opened the possibility to realize bosonic analogs of these states in platforms based on ultracold atoms and photons. Reaching the appropriate conditions for the simulation of the fractional quantum Hall effect with neutral particles (such as atoms and photons) has required decades of both theoretical and experimental efforts and passed through the implementation of many topological models at the single-particle level. However, we strongly believe that the stage is set finally and that bosonic fractional quantum Hall states will be realized soon in different set-ups. Motivated by this fact, we dedicate this Thesis to the study of the edge and quasihole excitations of bosonic fractional quantum Hall states with the goal of guiding near future experiments towards exciting discoveries such as the observation of anyons. In the first part of the Thesis we focus our attention on the behavior of the edge excitations of the bosonic $\nu=1/2$ Laughlin state (a paradigmatic wave function for the fractional quantum Hall effect) in the presence of cylindrically symmetric hard-wall confining potentials. With respect to electronic devices, atomic and photonic platforms offers indeed a more precise control on the external potential confining the systems, as confirmed by the recent realization of flat-bottomed traps for ultracold atoms and by the flexibility in designing optical cavities. At the same time, most of the theoretical works in this direction have considered harmonic confinements, for which the edge states have been found to display the standard chiral Luttinger liquid behavior, leaving the field open for our analysis of new physics beyond the Luttinger paradigm. In the second part we propose a novel method to probe the statistical properties of the quasihole excitations on top of a fractional quantum Hall state. As compared to the previous proposals, it does not rely on any form of interference and it has the undeniable advantage of requiring only the measurements of density-related observables. As we have already mentioned, although the existence of anyons have been theoretically predicted long time ago, it still lacks a clear-cut experimental evidence and this motivated people working with ultracold atoms and photons to push their systems into the fractional quantum Hall regime. However, while there exist plenty of proposals for the detection of anyons in solid-state systems (mostly based on interferometric schemes in which currents are injected into the system and anyons travel along its edges), in the present literature the number of detection schemes applicable in ultracold atomic and/or photonic set-ups is much smaller and they are typically as demanding as those proposed in the electronic context. Finally, in the last part of the Thesis we move to the lattice counterparts of the fractional quantum Hall states, the so-called fractional Chern insulators. Still with the purpose of paving the way for future experimental studies with quantum simulators, we focus our attention of the simplest bosonic version of these states and, in particular, on the properties of its quasihole excitations. Although this topic has already been the subject of intense studies, most of the previous works were limited either to system sizes which are too small to host anyonic excitations, or to unphysical conditions, such as periodic geometries and non-local Hamiltonians. Our study investigates for the first time the properties of genuine quasihole excitations in experimentally relevant situations.