This report deals with constrained coding in two dimensions. We describe the theory of constrained fields as a framework for addressing some of the challenges of code construction for advanced data storage devices that treat the recording media as a surface, rather than a series of tracks. The important concept of entropy is introduced. In general, the entropy of a constrained field is not readily computable, but we give a series of upper and lower bounds based on one dimensional techniques. We discuss the use of a Pickard probability model for constrained fields. The novelty lies in using a first order model to model higher order constraints by the use of an alphabet extension. We present an iterative method that based on a set of conditional probabilities can help in choosing the large numbers of parameters of the model in order to obtain a stationary model. Explicit results are given for the No Isolated Bits constraint. Finally we present a variation of the encoding scheme of bit-stuffing that is applicable to the class of checkerboard constrained fields. It is possible to calculate the entropy of the coding scheme thus obtaining lower bounds on the entropy of the fields considered. These lower bounds are very tight for the Run-Length limited fields. Explicit bounds are given for the diamond constrained field as well.