This thesis deals with problems related to the structure of the solutions to some specific polynomial equations. A brief introduction to the type of problems we are interested in is given in Chapter 1. In Chapter 2 we recall some standard results in number theory and additive combinatorics. In Chapter 3 we look at partition regularity of equations of the form xa + yb = zc over Z/pZ. In particular we look at the equation x+y= z2. In Chapter 4 we prove that any 2-colouring of N has infinitely many monochromatic solutions to the equation x + y = z². This work is joint with Ben Green. In Chapter 5 we use the same methods as in Chapter 4 to prove partition regularity of the equation x-y=z2. In Chapter 6 we show that a linear combination of kth powers is partition regular if and only if the corresponding linear equation is partition regular, provided the number of variables is large enough. This is based on joint work with Sam Chow and Sean Prendiville. In Chapter 7 we look at Heath-Brown's method of counting the zeros of a quadratic form in four variables, and in particular how the error term in this count is affected by the weight function used. In Chapter 8 we try to count the number of zeros of a quadratic form in four variables that lie in a fixed congruence class.