We now consider the theory of completeness and completions for quasi-uniform spaces. In Section 2 of this chapter we establish a satisfactory analogue of the completion theory of uniform spaces; indeed the results of Section 2 extend the usual theory. In this analogue, as in its uniform space counterpart, we have an ideal economy; every space has a completion and no space has two essentially different completions. The theory of completeness that we first consider has an oversupply of complete spaces and the reader who has become accustomed to the ideal economy of uniform completions may be dismayed by problems inherent in an imbalance of supply and demand. Some of the problems caused by the over-abundance of complete spaces are of interest (for example Problem C) and while the concept of completeness that we first consider does not allow an entirely adequate theory of completion it does provide a valuable link between topological properties and quasi-uniform properties.