This book puts together non-equilibrium thermodynamics, heat transport properties of superfluid He II, and thermodynamic and dynamic aspects of quantum turbulence. A one-fluid extended model of superfluid helium, with heat flux as an additional independent variable, is presented and compared with the two-fluid model, to explore how both models complement each other. Important features arise in rotating situations and in superfluid turbulence, characterized by quantized vortices leading to strong nonlinearities between heat flux and temperature gradient. The dynamics of vortex lines and their interaction with heat dynamics, a central topic in superfluid turbulence, is dealt with by introducing the vortex line density as an independent variable and writing its dynamical equations, considering the transitions from laminar to turbulent flows and from diffusive to ballistic regimes. Classical and quantum turbulence are compared from a mesoscopic view and from their energy spectra. The work also explores some parallelisms of quantum vortex thermodynamics with cosmic string thermodynamics and black-hole thermodynamics, exhibiting duality connections amongst them. It emphasizes didactical views over specialistic details, and may be used as an introduction to nonequilibrium thermodynamics of superfluid helium and its heat transport properties (second sound, nonlocal transport, nonlinear connections with quantum turbulence). The book is useful to researchers in superfluid helium, in heat transport, and in thermodynamics of cosmic strings and black holes. The diversity and complexity of its several physical equations will be inspiring for researchers in mathematical physics.