This is a report on a number of recent results on composition operators which map, for 0 <p⪯q∞, the Hardy spaceHp(on the unit disk in the complex plane) intoHq. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.