Consider a polynomial equation f(X,Y) in two variables (e.g. Y^6-5XY^5+X^3 Y^4-7X^2 Y^2+6X^3+X^4 = 0) . How do we solve for Y in terms of X, thus obtaining solutions to such an equation? In the talk, we will describe Newton polygons, work through example(s) (e.g. the one mentioned above whose solution Newton described) and outline Newton's method (a precursor to the Newton-Raphson method) of how to obtain these solutions as a Puiseux series. Time permitting, we will attempt at giving a rigorous explanation of why it works. For most part, the talk needs no specialised knowledge and should be widely accessible.