ABSTRACT. The question of when a metric projection has a continuous selection has received much attention in recent years. There has been a flurry of activity in this area by mathematicians from over the world. (At this writing, I am aware of relevant work by researchers located in over 17 different countries.) In spite of all this interest, of perhaps because of it, there are still many questions which remain unanswered. The writer gave a survey of continuous selections at a special meeting of the American Mathematical Society in January, 1982 (which appeared as [9]). At an Oberwolfach conference a few years later, Nurnberger and Sommer [22] concentrated their attention to the finite-dimensional subspaces in C[a, b] and gave a detailed description of the results available in this setting. Still later, the writer [10] announced some recent progress that had been made since the appearance of [9] and [22].

It is the purpose of my paper to announce the progress that has been made mainly in the last two years, and to point out some of the open problems which still remain. Many of these results have not yet appeared in print. In a sense, this report may be regarded as an updating and supplement to the papers [9], [10], and [22]. It is also hoped that this will provide additional motivation for others to take up these problems.