In this talk, we first motivate the use of singular terminal conditions: final trace of the solution of some semilinear PDEs or optimal control problems with constraints. Then we present some results concerning the existence of minimal supersolutions for a wide class of generators. Finally, we explain how the continuity problem (continuity of the solution at the terminal time) can be solved in some specific cases. This talk is based on several papers with T. Kruse, D. Marushkevych and D. Sezer.