Several approximation operators followed Philippe Clément’s seminal paper in 1975 and are hence known as Clément-type interpolation operators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space *V* ⊂ *W*^{k,p}(Ω) onto some finite element space *V*_{h} ⊂ *W*^{k,p}(Ω) and generalize nodal interpolation operators whenever *W*^{k,p}(Ω) ⊄ *C*^{0}(Ω), i.e., when *p* ≤ *n/k* for a bounded Lipschitz domain Ω ⊂ ℝ^{n}. The original motivation was *H*^{2} ⊄ *C*^{0}(Ω) for higher dimensions *n* ≥ 4 and hence nodal interpolation is not well defined.

Todays main use of the approximation operators is for a reliability proof in a posteriori error control. The survey reports on the class of Clément type interpolation operators, its use in a posteriori finite element error control and for coarsening in adaptive mesh design.