In this thesis, we study the mechanics of tight physical knots. Knots are omnipresent in surgery, climbing, and sailing, with disastrous consequences when the filament or the rope fails to perform its function. Even if the importance of mechanical analysis of knots has long been recognized, established guidelines for best practices typically rely primarily on empirical data gained from historical experience, not structural analysis. Existing models in knot theory or Kirchhoff's theory for elastic rods are often insufficient to describe the functional behavior of tight physical knots due to the intricate three-dimensional nonlinear geometries, large elastic (and sometimes plastic) deformations, and frictional interactions. A few past studies reported stability and knot quality measurements, which, however, are specific and difficult to generalize. We tackle the study of the mechanics of physical knots by combining topological and geometric arguments, precision-model experiments, and high-fidelity numerical simulations using the finite element method. An elaborate toolbox is developed for volumetric imaging of X-ray micro-computed tomography data. Our investigation is centered on the shape of physical trefoil knots, the performance of stopper knots, and the strength of surgical knots. First, we perform a compare-and-contrast investigation between the equilibrium shapes of physical and ideal trefoil knots, in closed and open configurations. We construct physical realizations of tight trefoil knots tied in an elastomeric rod. X-ray tomography and 3D finite element simulation allow for evaluating the role of elasticity in dictating the physical knot's overall shape, self-contact regions, curvature profile, and cross-section deformation. The results suggest that regions of localized elastic deformation, not captured by the geometric models, act as precursors for the weak spots that could compromise the strength of knotted filaments. Second, we investigate the performance of stopper knots, which prevent the rod end from retracing through a narrow passage. We develop a physical model involving an inextensible elastomeric rod, onto which a figure-eight knot is tied and pulled against a rigid stopper plate. A complex interplay of frictional interactions and friction-induced twist leads to capsizing, a mechanism that rearranges the knot configuration while keeping its topology. In contrast to isotropic rods, we find that the decoupling of bending and twisting rigidities in rope-like structures penalizes rod twisting, which impedes the knot from capsizing. Third, we study the operational and safety limits of surgical sliding knots, highlighting the previously overlooked but crucial effect of plastic deformation. The relevant range of applied tensions, geometric features, and the resulting knot strengths are characterized using experimental and numerical model systems. Finally, we find that all the experimental and numerical data, involving all the knot configurations we investigated and a wide range of friction coefficients, collapse onto a master curve. The acquisition and study of unprecedented experimental data, combined with FEM simulations, has enabled us to systematically explore the different ingredients dictating the mechanical performance of knotted structures with highly nonlinear geometrical features and material properties.