Skeletal muscles are a biological tissue with unique mechanical properties, characterised by anisotropic effects determined by its fiber-like microstructure. It features a nonlinearly elastic response and activation phenomena. Of particular relevance is also the multiscale nature of the muscle tissue. Indeed, sarcomeres, the smallest contractile units, are bundled to form miofibrils, which in turn constitute the muscle in its large-scale configuration. Experimental evidence indicates that the mechanical response of the smaller units is rather different from and simpler than what can be measured for the whole muscle. Hence, a multiscale approach is necessary to arrive at both an understanding and a proper continuum modelling of muscle dynamics. We first present a theoretical framework for anisotropic nonlinear elasticity based on the decomposition of strain and stress tensors on a tensorial basis adapted to the local anisotropy of the material. The presence of local anisotropies is reflected on material symmetries, and we consider the corresponding restrictions for transversely isotropic materials. This formalism aims at a clear and mechanically motivated organisation of the degrees of freedom involved in describing nonlinear elasticity, to facilitate the experimental identification of material functions for their constitutive characterisation. Within the said framework, we start from experimental data about the passive response of muscle specimens to identify nonlinear material functions for the stress-strain relation, thereby following a data-driven approach. The peculiar nonlinearity of the response can be ascribed to the microscopic heterogeneity that leads to a progressive recruitment of different fibers. We propose a basic model in the context of anisotropic Cauchy elasticity to capture these phenomena and then propose a strategy to include an activation mechanism. Finally, it is important to consider the significant sources of dissipation that affect the muscular dynamics. For this reason, we explore how to include in the constitutive law terms that produce a viscous damping during the passive and active motion.