Problems and procedures in assessing and obtaining fit of data to the Rasch model are treated and assumptions embodied in the Rasch model are made explicit. It is concluded that statistical tests are needed which are sensitive to deviations so that more than one item parameter would be needed for each item, and more than one person parameter would be needed for each person. Statistical goodness-of-fit tests--based on the conditional maximum likelihood estimates of the item parameters--which can detect these two kinds of deviation are presented. Common sources of deviation are also identified, as are the tests needed to detect them. Problems in the use of statistical tests to assess fit are discussed and some investigations of power are presented. In relation to a distinction between use of the Rasch model as a criterion and as an instrument, the treatment of the goodness-of-fit problem in different measurement contexts is discussed. Finally, it is concluded that items which can be identified as misfitting should not be routinely excluded to obtain fit to the model; instead other actions should often be taken--such as grouping of the items into homogeneous subsets. (Author/CP)