The conservation laws for (3+1)-dimensional Calogero-Bogoyavlenskii Schiff (CBS) equation are derived via Noether’s approach associated with the Standard Lagrangian. The classical Lie point symmetries are computed for CBS equation and the symmetry conservation laws relation is employed which shows that which symmetry is associated with the conservation law. The double reduction theory is used to reduce the number of independent variables and order of CBS equation via similarity variables. The inverse similarity transformations give rise to exact solutions of CBS equation. In addition to the above, different combination of Lie symmetries are used to derive several independent solutions. The analysis will be extended to class of partial differential equations involving arbitrary functions. Moreover, we shall compute symmetries, conservation laws and exact solutions of Boussinesq-type equation, Burger equation, (2+1)-dimensional Zoomeron equation and Timoshenko-Prescott form of Beam equation. This study points out new ways of finding exact solutions as well as reduction in number of independent variables for partial differential equations arise in different areas.