Last year we got a small black cat (a tomcat actually) that was so active and wanted to get outside to play, it drove us crazy to keep him inside. But always opening and closing the door of our flat was too annoying. We took a look at the generic cat doors that you build into the door or window, but all the solutions were (including the destruction of the window or flat door) much too expensive.
However, we have a bathroom with a pivot-hung window that leads to the stairs. So we built a setup where the cat jumps on a small ladder that leads to the top of the window frame. On top of the window frame is a box with a flap made of carpet. The cat can push the flap open and step through the hole on a small platform on the other side of the window. From there it's just a small jump to the ground. The way back in functions similar, only in the opposite direction.
The wooden construction covers all the sides of the (now always) open pivot-hung window without destroying the window itself. If we change the flat someday, we can just remove the cover and the window will be as it was before. The carpet flap ensures that not too much heat gets lost during the winter and not too many bugs enter the flat during the summer.
All in all the construction used only a bit of wood and time and is in use for a bit more than a year now still working like a charm. If my description of the process was not accurate enough you can watch the following video my girlfriend made of our cat "Socke" (means sock in swiss-german, for he has one white paw..) leaving the flat. Hope you enjoyed the principle and maybe it helps somebody with a similiar cat-problem. :)