Nanomechanical resonators have extremely low masses (~ 10−15 kg) and frequencies from a few megahertz all the way up to the gigahertz range. These properties along with a small damping rate make them very useful or ultrasensitive detection applications, now pushing into the realm of zeptonewtons (10−21 N) and zeptograms (10−21 g). On a more fundamental level, nanomechanical resonators are expected to display quantum mechanical effects when cooled down to millikelvin temperatures. The understanding of dissipation in nanomechanical resonators is important for device applications and to study quantum mechanical effects in such systems. However, despite a range of experiments on semiconducting and metallic devices, dissipation in nanomechanical resonators at low temperatures is not yet well understood. Although mechanical resonators have traditionally been operated in the linear regime, exploiting their nonlinearities can prove advantageous for industrial applications as well as opening up new experimental windows into the fundamental study of the nonlinear dynamics of mesoscopic systems. In this thesis, we present results from low temperature dissipation studies on pure gold and on gold-coated high-stress silicon nitride nanomechanical resonators. A theory, which predicts the existence of tunnelling two-level systems (TLS) in bulk disordered solids at low temperatures, is used as a framework to describe the data. The nonlinear interactions between different flexural modes of a single silicon nitride device, are explored experimentally and theoretically. The resonators were fabricated as doubly-clamped beams using a combination of optical lithography, electron-beam lithography, dry and wet etching techniques. The motion of the resonators was actuated and detected using the magnetomotive scheme. At low temperatures, all the beams had resonant frequencies between 3 and 60 MHz and quality factors in the range 105 − 106. The strong variation observed in dissipation and resonant frequency at the lowest temperatures (below 1 K) indicates the presence of tunnelling TLS in nanomechanical resonators.