Quantum Information Theory is about studying information encoded in quantum systems and exploring their unique properties. The topic of our report is Quantum Data Compression, a central problem to storage and transmission of data. Our report presents an overview of Classical Coding Theory and the main concepts of Quantum Coding Theory. Moreover, we introduce a particular quantum noiseless and lossless data compression scheme. Finally, we present three methods to improve this scheme. Our first method, Brute Force (BF), presents only significant advantages for small amounts of data, but has otherwise a very high complexity time. This is further improved with our scheme Improved BF, where the complexity time is decreased by taking profit of the advantages of quantum algorithms. Another alternative is our Adapted Algorithm, based on a different way of ordering probabilities and which offers the best complexity time, but contrary to the previous two methods does not guaranty that an optimal solution will be found.