Let A be a set containing at least two elements and let n be a positive integer. Let \(O_ A^{(n)}\) \((P_ A^{(n)})\) denote the set of all (partial) n-ary operations on A and put \(O_ A:=\cup \{O_ A^{(m)}|\) \(m\in {\mathbb{N}}\}\) and \(P_ A:=\cup \{P_ A^{(m)}|\) \(m\in {\mathbb{N}}\}\). A clone on A is a subset of \(O_ A\) which is closed under composition and contains all projections. Let \(\infty\) be an element outside of A and put \(B:=A\cup \{\infty \}\). Let \(R_ B^{(n)}\) denote the set of all \(f\in O_ B^{(n)}\) with \(f(\infty,...,\infty)=\infty\) satisfying the following condition: If \(i\in \{1,...,n\}\) and if f depends on its i-th variable then \(f(x_ 1,...,x_{i-1},\infty,x_{i+1},...,x_ n)=\infty\) for all \(x_ 1,...,x_{i-1},x_{i+1},...,x_ n\in B\). Put \(R_ B:=\cup \{R_ B^{(m)}|\) \(m\in {\mathbb{N}}\}\). Then \(R_ B\) is a clone on B. For all \(f\in R_ B^{(n)}\) let \(\phi\) (f) denote the restriction of f to \(\{(x_ 1,...,x_ n)\in A^ n|\) \(f(x_ 1,...,x_ n)\in A\}\). A partial clone is an image of some subclone of \(R_ B\) under \(\phi\). Let \(p_ n\) denote the partial n-ary operation on A with empty domain. Then \(O_ A\cup \{p_ m|\) \(m\in {\mathbb{N}}\}\) is the unique partial clone C with \(O_ A\subsetneqq C\subsetneqq P_ A\). For finite A the completeness of a subset of \(P_ A\) is characterized.