This seminar on number-theoretical algorithms and their applications was the sixth on this topic at Dagstuhl over a period of seventeen years. This time 39 people from 10 countries participated.

One of the major goals of these seminars has been to broaden interactions between number theory and other areas. For instance, there has been an effort to bring together people developing the theory of efficient algorithms with people actually writing software. There has also been continuing interest in cryptography. These aspects were both emphasized by the topics of special interest in this year: Number Theoretical Software and Algorithms for the Post Quantum Era.

About half of the 24 talks given were in these areas showing rapidly growing interest. One fourth of the talks were on curves, most with an eye to applications in cryptography.

The other talks focused on more classical topics of algorithmic algebraic number theory. We just mention the calculation of global fields and of class groups.

Even though we had less participants than at the last meeting the group seemed to be more homogenous. The variety of topics of the talks was stimulating to the audience. Their smaller number gave more room for discussions. It did not come as a surprise that these were most intensive in our emphasized topics.

For example, number theoretical software was not only discussed but also developed during the meeting. The participants did indeed a lot of coding. We would like to mention hat M. Stoll has a C program called ratpoints which is very fast at finding rational solutions to $y^2=f(x)$. During the conference this program was incorporated into Sage, a process which included finding several bugs (memory leaks) in ratpoints so that M. Stoll could fix them right there.

The reaction of the participants was very positive and we believe that we succeeded in having an effective meeting that was able to appeal to a broad audience. We made sure to allow for adequate breaks between sessions, and - as already mentioned - there were many opportunities for discussions that the participants took advantage of. The pleasant atmosphere of Schloss Dagstuhl once again contributed to a very productive meeting. Even more positively, several younger people who were there (for the first time) told us that they not only found it a very good meeting indeed, but the best venue they had been to for a conference.