Dark Energy, the mechanism driving the cosmic late-time acceleration, has remained a mystery since its discovery almost two decades ago. In the current standard model, ΛCDM, Dark Energy is modelled by adding the energy density of vacuum via the the cosmological constant, Λ. However, this scenario requires a fine-tuning on the level of several orders of magnitudes. One alternative explanation to the Cosmological Constant could be that the laws of General Relativity (GR) break down on the largest scales of the Universe; potential modifications to GR could give rise to terms or mechanisms that drive and explain the observed late- time expansion without the need to include highly fine-tuned parameters. One of these alternative theories are Covariant Galileons: a theory of gravity introducing a new scalar degree of freedom whose kinematics source an accelerated expansion of the Universe at late times. This thesis reviews how all three branches – cubic, quartic and quintic – of the minimally coupled Covariant Galileon can be ruled out by combining and cross-correlating different cosmological and astrophysical probes. Besides the observations of the Cosmic Microwave Background and Baryonic Acoustic Oscillations, we focus on measurements of the Integrated Sachs-Wolfe (ISW) effect and multi-messenger astronomy through gravitational wave (GW) observations and the associated electromagnetic counterparts. It is shown that the cubic model exhibits a 7.8σ tension with measurements of the ISW effect. The parameter space of quartic and quintic models can only be constrained by this observation, but can not be completely invalidated. However, Quartic and Quintic Galileons predict an anomalous propagation speed of tensor perturbations, which is inconsistent with the short delay in arrival time of GW170817 and the associated gamma-ray burst GRB170817A. With the combined constraints arising from the ISW effect and the propagation speed of GWs, the cosmological viability of minimally coupled Covariant Galileons is invalidated. The methods presented here rely on fairly generic properties of modified gravity theories. Hence, they are also expected to set constraints on other alternative theories of gravity to narrow down the possible explanations for Dark Energy.