Abstract: This study is a survey of existing fractal coding methods and approaches, which highlights the solution to the key and open issues in the fractal coding. In addition, the possible approaches from encoding time optimization point of view are discussed. This study can be useful for the beginners in the fractal image compression for gray and color images. Neural network based approaches are also summarized. A fragment of neural network architecture is given to develop the neural network based approach for fractal gray and color image compression. This study may help for the implementation of straight fractal coding and neural network based approach.