Improvements in our knowledge of the absolute value of the Newtonian gravitational constant, G, have come very slowly over the years. Most other constants of nature are known (and some even predictable) to parts per billion, or parts per million at worst. However, G stands mysteriously alone, its history being that of a quantity which is extremely difficult to measure and which remains virtually isolated from the theoretical structure of the rest of physics. Several attempts aimed at changing this situation are now underway, but the most recent experimental results have once again produced conflicting values of G and, in spite of some progress and much interest, there remains to date no universally accepted way of predicting its absolute value. The review will assess the role of G in physics, examine the status of attempts to derive its value and provide an overview of the experimental efforts that are directed at increasing the accuracy of its determination. Regarding the latter, emphasis will be placed on describing the instrumentational aspects of the experimental work. Related topics that are also discussed include the search for temporal variation of G and recent investigations of possible anomalous gravitational effects that lie outside of presently accepted theories.