Problems of set-theoretic non-standard analysis

The main results in the area of set-theoretic non-standard analysis (non-standard set and class theories) obtained over the last few years are presented. It is demonstrated how a universe of a comparably simple theory (beginning with the usual Zermelo-Fraenkel set theory ZFC) can be extended to a universe of a more complicated non-standard set or class theory. The last section develops the foundations of Boolean-valued analysis as a part of set-theoretic non-standard analysis.