opological physics is emerging as an active area of research, addressing fundamental questions on how geometry, symmetry, and topology affect physical properties, paving the way toward novel technological applications. Originally investigated in quantum systems, these concepts have been thereafter translated across diverse domains including, electromagnetic, plasmonic, elastic, and acoustic waves. Specifically, in elasticity, due to the strong tendency to hybridize of wave modes with different polarization, topological protection is viewed as a revolutionizing approach to design waveguides supporting unique features such as (i) being immune to defects and (ii) suppressing backscattering during the wave propagation phenomenon. These novel features arise as a consequence of their dispersion surface topology. This Tutorial aims to introduce the theoretical, numerical, and experimental frameworks to investigate topological elastic waveguides, discussing the key ideas, first, in the context of discrete systems, and then, in continuous elastic solids. After a comprehensive description of the currently used state of the art scientific techniques, various classes of topological wave phenomena leading to localized waves in elastic architected plates and beams are presented. Implications of the presence of both longitudinal and shear waves in elastic solids are discussed, and the associated challenges, opportunities, and strategies to exploit their interplay highlighted. The symmetry conditions required to induce them are discussed using a number of representative examples. Finally, future research directions of this fledgling field are outlined.