A selection of recent results and problems in Cp-theory

The author presents five popular topics in Cp-theory giving their general description together with some unsolved problems. The main idea is to show the areas of major activities displaying what is being done nowadays and trying to outline the trends of their future development. A big percentage of the cited results and problems are new, i.e., published/obtained in the 21-st century. However, some classical old theorems and questions are also cited the author being convinced that they are worth to be repeated with a new emphasis due to what is achieved at the present time.