Calculations in exceptional groups over rings

In the present paper, we discuss a major project whose goal is to develop theoretical background and working algorithms for calculations in exceptional Chevalley groups over commutative rings. We recall some basic facts concerning calculations in groups over fields, and indicate complications arising in the ring case. Elementary calculations as such are no longer conclusive. We describe the basics of calculations with elements of exceptional groups in their minimal representations, which allow one to reduce calculations in the group itself to calculations in subgroups of smaller rank. For all practical purposes, such calculations are much more efficient than localization methods. Bibliography: 147 titles.