It is very difficult in general to determine when a given compact in C^n, n>1, is polynomially convex. In this talk, we shall discuss polynomial convexity of some classes of sets. First, we shall consider two totally-real surfaces in C^2 that contain the origin and have distinct tangent planes there. We shall discuss how the local polynomial convexity of the union of the tangent planes at (0,0) influences local polynomial convexity of the union of the surfaces at (0,0). Secondly, we will present a condition for local polynomial convexity of unions of more than two totally-real planes in C^2 containing the origin. Next, we shall talk about pluri- subharmonicity. Using this notion we shall give a new proof of an approximation theorem of Axler and Shields and also generalize it. Polynomial convexity plays a very central role in our proof. Finally we shall discuss a characterization for (large) compact patches of smooth totally-real graphs in C^{2n} to be polynomially convex.