This paper presents a survey of different methods in the prospect of medical and brain image compression. Compressing an image is basically a process of encoding the image to reduce the size of image as a number of bytes for storage and transmission purposes. This should be done while preserving as much as possible the quality of the image. Hence, image compression has been founded necessary in different medical imaging systems, where there are too much number of bytes of the reconstructed images. There are basically two categories of compression techniques; namely lossless and lossy compression techniques. As the name indicates, in the lossless technique the image is encoded without any loss of information.But in the lossy one, it is allowed that some information is missed. The lossy compression techniques are commonly applied to multimedia data such as audio, video, and still images. Several lossless and lossy compression approaches have been applied to medical images. Different compression approaches belonging to the two categories are discussed and Brain images compression techniques are highlighted. Furthermore, quantitative comparisons between different compression methods are given.as well as advantages and disadvantages of each method.