In the present thesis, “Hydrodynamics of Binary Bose-Einstein Condensates and Hydro-elasticity of the Inner Crust of Neutron Stars”, the hydrodynamic effects, instabilities and superfluid turbulence in binary immiscible ultracold gases, and hydro-elastic macroscopic coupled modes and microscopic structure of the inner layers of the crust of neutron stars, are studied. The ultracold gas dynamics can be realized in the laboratory. The excitation modes of the inner crust determine a number of observable properties such as elasticity, thermal properties and mass transport properties. Here we focus on expanding the details, rather than repeating the results presented in the published articles. In the part of the thesis related to atomic ultracold gases, we utilize the physical parameters in the experimentally realizable parameter region. We numerically simulate the coupled non-linear Schrödinger equations, and calculate observable quantities, such as phase and modulus of the order parameter, conditions needed for observation of the Rayleigh-Taylor instability and for turbulence generation. The numerical calculations are accompanied by analytical description of the processes. The dispersion relation for capillary-gravitational waves at the interface between two ultracold gases, is derived straightforwardly from the superfluid Lagrangian. The equations of motion for centre-of-mass of the superfluids are derived, and then used in our model of the quantum swapping of immiscible superfluids pressed by a strong external force. By numerical simulation, we find that the Kelvin-Helmholtz instability which occurs at the non-linear stage of the Rayleigh-Taylor instability, can generate quantum turbulence with peculiar properties. We find that two-dimensional superfluid systems with weak inter-component repulsion are different from previously studied strongly repulsive binary superfluids, because the quantum Kelvin-Helmholtz instability in weakly repulsive superfluids rolls up the whole interface forming a vortex bundle, similarly to dynamics of the shear fluid layers in the classical hydrodynamics. Production of vortex bundles favours the Kolmogorov spectrum of turbulence, and we find that the Kolmogorov scaling indeed is present in a freely decaying turbulence. In the part of the thesis related to neutron stars, we study the inner crust of neutron stars, where the fully ionized atomic nuclei coexist with a superfluid of neutrons. The interaction between superfluid neutrons and the crystallized Coulomb plasma is due to the interaction between density perturbations (interaction of the scalar type), and between the current - the non-dissipative entrainment effect (interaction of the vector type). We calculate velocities of the collective modes of the crystal coupled to superfluid neutrons. As an input we use the results of microscopic nuclear calculations in the framework of the compressible liquid drop model (the Lattimer and Swesty equation of state), and more recent effective Thomas-Fermi calculations with shell corrections (N. Chamel, and the Brussels theoretical nuclear physics group). Knowledge of velocities as functions of the matter density in the inner crust is important for calculation of a number of dynamic and transport properties. The heat transport properties of the inner crust are directly observable in accreting binary systems (low-mass x-ray binaries). The mass transport properties of the inner crust are directly linked to the rotational evolution, being a key physical ingredient of the pulsar glitch phenomenon. The elastic properties are related to the vibrational modes of the star, and to the breaking stress of the crust. In the second part of our work on neutron stars we investigate the microscopic structure of the inner crust treating the structure as an anisotropic crystal coupled to s-wave superfluid neutron liquid. As we analyse dynamics of the elementary excitations at higher wavenumbers (smaller scales), we reach the edge of the first Brillouin zone. The Lattimer-Swesty data is applicable for wavenumbers much smaller than the edge of the first Brillouin zone. We extrapolate the data through the whole first Brillouin zone to calculate the fastest growth rate of the unstable modes. The crucial step is to calculate the mode velocities in anisotropic crystal incorporating both the induced neutron-proton interactions, and the electron screening properties. We find that the combined influence of these two effects leads to softening of the longitudinal phonon of the lattice above about the Thomas-Fermi screening wavenumber of the electrons. The critical wavenumber when the frequency becomes purely imaginary is about  1/5 - 2/3  of the reciprocal lattice vector, thus validating our assumption. The imaginary mode frequency implies instability at finite wavenumbers. Our calculations suggest that the mode at the first Brillouin zone edge is the most unstable, and thus the structure experiences a displacive phase transition when the central ion of a unit cell of the body-cubic-centred lattice, is displaced to the cube face. Thus, the electronic structure of matter at densities above the neutron drip [1], is richer than previously appreciated, and new microscopic calculations of nuclear structure are necessary which take into account the high-wavenumber physics. Such calculations will provide crucial input to models interpreting the quasi-periodic oscillations in Soft Gamma Repeaters as magnetar x-ray flares, and to the theory of glitches of neutron stars. [1] The neutron drip density is ~3×1011 g cm-3.