Menger [4] initiated the study of probabilistic metric spaces in 1942. A probabilistic metric space (briefly a PM space) is a space in which the "distance" between any two points is a probability distribution function. These spaces are assumed to satisfy axioms which are quite similar to the axioms satisfied in an ordinary metric space. The triangle inequality has been the subject of some controversy. Menger's triangle inequality was first challenged by Wald [16] who suggested an elegant form of the triangle inequality to replace Menger's. In [-7] Schweizer and Sklar provided sufficient reason to indicate that Wald's inequality was restrictive and returned to Menger's triangle inequality which they modified slightly. The subject then began to grow rapidly due to the work of Schweizer, Sklar, Thorp and others. In [10, 11] Serstnev introduced yet another form of the triangle inequality. His formulation includes Wald's and Menger's formulation but for a slight exception which at present seems to be uninteresting. For this reason we shall state and prove our results in the setting of Serstnev whenever it is appropriate. In Section 2 we shall do two things. First we shall briefly take note of the fact that the principal result of our paper on completions of PM spaces can be obtained under weaker hypotheses. Then we shall answer the following consistency questions: (1) Is a complete metric space obtained when a complete PM space is metrized? and (2) If a PM space and its completion are metrized, will the completions of the resulting metric spaces be isometric? In Section 3 contraction maps on PM spaces will be investigated. A very natural definition of a contraction map was introduced by Sehgal [-9]. In that paper he showed that every contraction map on a complete PM space satisfying the strongest form of Menger's triangle inequality has a unique fixed point. We shall give a strong plausibility argument which will indicate that this result is the exception rather than the rule for PM spaces. We shall prove that it is possible to construct complete PM spaces together with contraction maps which have no fixed points. In fact this will be done so that any of a very large class of triangle inequalities is satisfied. Finally we shall utilize the additional structure of an E-space to define a stronger contraction map which will have a fixed point whenever the space on which it is defined is complete. In the last section the analogues for complete PM spaces of two other classical theorems for complete metric spaces will be proved. A few definitions and conventions will be made here to fill in some background for the reader.