The usability of Mobile devices and portable computers has been increased very rapidly. The main focus of modern research is to handle the big data in Mobile and pervasive computing. This paper gives an overview of Mobile Data Challenge which was a smart phone based research done by Nokia through Lausanne Data Collection Campaign. The Mobile Data Challenge was introduced when the amount of mobile data was seen to be increased very much in a short period of time. The rise of big data demands that this data can be accessed from everywhere and anytime. For handling such a huge amount of data e.g. SMS, images, videos etc, this data must be compressed for easy transmission over the network. It also helps in reducing the storage requirements. There are various techniques for the compression of data that are discussed in this paper. SMAZ and ShortBWT techniques are used to compress SMS. JPEG and Anamorphic Stretch Transform are used to compress the images.