The term 'running score' refers to a TALLY of points or other measurements that are kept up-to-date throughout a game or other event. It is often used in sports, such as baseball, where the score is kept track of inning by inning. The term can also be used in other contexts, such as keeping track of the number of questions answered correctly in a quiz or the amount of money won or lost in a gambling game. The crossword solution for this term is 'TALLY, ' which is a verb meaning to count or keep track of something.