Amorphous solids are structurally disordered. They are very common and include glasses, colloids, and granular materials, but are far less understood than crystalline solids. Key aspects of these materials are controlled by the presence of excitations in which a group of particles rearranges. This motion can be triggered by (a) quantum fluctuations associated with two-level systems (TLS), which dominate the low temperature properties of conventional glasses and have practical importance on superconducting qubits; by (b) thermal fluctuations associated with activations, which are related to the famous and challenging glass transition'' problem; or by (c) exerting an external stress or strain associated with shear transformations, which control the plasticity. Hence, it is important to understand how temperature and system preparation determines the density and geometry of these excitations. The possible unification of these excitations into a common description is also a fundamental problem. These local excitations are thought to have a close relationship with Quasi-localised modes (QLMs)'' which are present in the low-frequency vibrational spectrum in amorphous solids. Understanding the properties of QLMs and clarifying the relation between QLMs and these local excitations are important to the study of the latter. In this thesis: (1) we provide a theory for the QLMs, D_L(omega) ~ omega^alpha, that establishes the link between QLMs and shear transformations for systems under quasi-static loading. It predicts two regimes depending on the density of shear transformations P(x)~ x^theta (with x the additional stress needed to trigger a shear transformation). If theta>1/4, alpha=4 and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If theta<1/4, alpha=3+ 4 theta and all QLMs form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically. (2) We present a protocol to generate extremely stable computer glasses at minimal computational cost. It consists of an instantaneous quench in an augmented potential energy landscape, with particle radii as additional degrees of freedom. (3) We propose a unification of theories predicting a gap in the spectrum of QLMs of the Hessian (Stiffness Matrix) that grows upon cooling, with others predict a pseudo-gap D_L(omega)} ~ omega^alpha. Specifically, we generate glassy configurations of controlled gap magnitude omega_c at temperature T=0, using so-called `breathing' particles, and study how such gapped states respond to thermal fluctuations. We propose an interpretation of mean-field theories of the glass transition, in which the modes beyond the gap act as an excitation reservoir, from which a pseudo-gap distribution is populated with its magnitude rapidly decreasing at lower T. (4) Preliminary results on the local excitations are presented for glasses (realistically prepared glasses) obtained by an instantaneous regular quench from the equilibrated configurations at low temperature T_p obtained by the SWAP Monte Carlo method. (5) The relationship between the density of TLS and the density of QLMs is built up.