Computer aided algebraic geometry: constructing surfaces of genus zero

Everybody knows that mathematics has a key role in the development of the modern technology. It is less known that the modern technology gives something back to mathematics. In this note we give an account on how the combination of classical results as the Riemann Existence Theorem with the use of computers and computational algebra programs answered interesting old-standing problems in classical algebraic geometry, namely regarding the construction and the classification of new surfaces of general type. We also give a full list of the surfaces constructed with this method up to now, and present the next challenges on the subject.