Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers. This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This ``quantum parallelism'' outstrips by far any parallelism that can be thought of in classical computation and is responsible for the ``exponential'' speed-up of computation. This is a non-technical (or at least not too technical) introduction to the field of quantum computation. It does not cover very recent topics, such as error-correction.