Kinetic equations occur in mesoscopic models for many physical phenomena. The direct solution of the kinetic equation is prohibitively expensive due to the high dimensionality of the space of independent variables. A viable alternative is to reformulate the problem in terms of the moments of the distribution function. Recently, a suite of quadrature-based moment methods has been developed for approximating solutions to kinetic equations. This suite of quadrature-based moment methods has several desirable properties that makes it more efficient and robust compared to the other moment methods. Despite these desirable properties, there is often a bottleneck associated with these methods. Use of higher than first-order discretization schemes for the convection terms and higher than second-order discretization schemes for the diffusion terms often leads to non-realizable moment sets. A non-realizable moment set does not correspond to a non-negative distribution function. The discretization schemes that can guarantee the non-negativity of the distribution function are called realizable schemes. The standard high-order discretization schemes are non-realizable. As a part of current research study, a set of high-order realizable schemes has been developed for both convection and diffusion terms that guarantee the non-negativity of the distribution function using constraints on the time step size, known as realizability conditions. In addition, the current study also shows the application of quadrature-based moment methods to two multi-physics phenomena - bubble-column flow and radiation transport. The two problems have been formulated and solved using the quadrature-based moment methods with particular attention to realizability.