We discuss the problem of embedding a PTS\((\text{n},\lambda)\) (a partial triple system on \(n\) vertices with index \(\lambda\)) in a TS\((\text{n},\lambda)\) (a triple system with \(n\) vertices and index \(\lambda\)) whenever \(t\) is admissible and \(t \leq 2n+1\). We bring out the close connection between this problem and various edge-colouring problems. The work described is mostly due to the author and C.A. Rodger.