We discuss the fluctuating hydrodynamic description used to study the behavior of a system of passive particles and that of active matter, comprising self-propelled particles. The dynamics are primarily formulated in terms of a set of collective modes {ψ} of the system. We will discuss how, starting from a set of microscopic balance equations for these modes, the description of a system with smooth Spatio-temporal dependencies follows. The breaking of Galilean invariances in the equations for active-matter hydrodynamics is linked to the microscopic dynamics of the individual units. We also discuss the equations of fluctuating hydrodynamics for many-particle systems whose microscopic units have both translational and rotational motion. The orientational dynamics of each element are studied in terms of Langevin equations for the rotational motion of a corresponding fixed-length director u with an individual unit. For the Brownian dynamics, only in terms of position variables, noise in the Langevin equation for the director u is multiplicative. The form of the corresponding coarse-grained equation for the so-called Iˆto and Stratonvich interpretation of the multiplicative noise in the u-equation is different. The coarse-grained equations of hydrodynamics are obtained by averaging with respect to a local equilibrium distribution, which involves an extended set of dynamical variables for the rotational motion. From the stationary solution of the (deterministic) equation for the probability distribution P[ψ], we obtain a free energy functional F[ψ]. The F[ψ]s for the different FNH descriptions with their corresponding set of {ψ} are worked out. The applications of the coarse-grained equations on the long-time dynamics will be discussed.