We will study the algorithmic aspects of quantum information. Our goal is to find new algorithms for quantum computers and new quantum communication protocols that are more efficient than the classical protocols. We will pursue this goal in a number of different ways.Firstly, we will study concrete problems that may have fast quantum algorithms and develop methods for constructing quantum algorithms. We will investigate the use of quantum walks (quantum counterparts of random walks) and learning graphs (a very new method which appeared less than a year ago) to construct quantum algorithms. We will also study two promising application areas for new quantum algorithms: hidden shift problems and property testing.Secondly, we will investigate general properties of quantum algorithms, such as the role of various resources (e.g., quantum entanglement or quantum discord) in quantum algorithms. We will study restricted forms of quantum computation, to find the minimum conditions under which universal quantum computation is possible. We will investigate the role of structure in input data in quantum speedups, by studying the maximum quantum speedups achievable in various settings.Thirdly, we will study the counterparts of those questions in the communication setting. We will work on designing quantum protocols that solve communication tasks more efficiently than any classical protocol and investigate quantum communication in a game-theoretic setting where parties act to maximize their self-interest.Lastly, we will investigate applications of quantum information concepts in other areas, namely to solve classical problems in computer science and to understand the computational complexity of problems in quantum physics. One of the major applications of a future quantum computer is to simulate quantum physical systems. Our goal is to understand this area in terms of complexity and possible quantum algorithms and to compare these with classical computational techniques.