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When human culture advances, current problems in science and engineering become more complicated and need more computing power to tackle and analyze. A supercomputer is not the only choice for complex problems any more as a result of the speed-up of personal computers and networks. Grid technology, which connects a number of personal computers with high speed networks, can achieve the same computing power as a supercomputer does, also with a lower cost. However, grid is a heterogeneous system. Scheduling independent tasks on it is more complicated. In order to utilize the power of grid completely, we need an efficient job scheduling algorithm to assign jobs to resources in a grid. In this paper, we propose an Adaptive Scoring Job Scheduling algorithm (ASJS) for the grid environment. Compared to other methods, it can decrease the completion time of all submitted jobs, which may compose of computing intensive jobs and data-intensive jobs.