Computational Grids are evolving into a global, service-oriented architecture – a universal platform for delivering future computational services to a range of applications of varying complexity and resource requirements. The thesis focuses on developing a new scheduling model for general-purpose, utility clusters based on the concept of user requested job completion deadlines. In such a system, a user would be able to request each job to finish by a certain deadline, and possibly to a certain monetary cost. Implementing deadline scheduling is dependent on the ability to predict the execution time of each queued job, and on an adaptive scheduling algorithm able to use those predictions to maximise deadline adherence. The thesis proposes novel solutions to these two problems and documents their implementation in a largely autonomous and self-managing way. The starting point of the work is an extensive analysis of a representative Grid workload revealing consistent workflow patterns, usage cycles and correlations between the execution times of jobs and its properties commonly collected by the Grid middleware for accounting purposes. An automated approach is proposed to identify these dependencies and use them to partition the highly variable workload into subsets of more consistent and predictable behaviour. A range of time-series forecasting models, applied in this context for the first time, were used to model the job execution times as a function of their historical behaviour and associated properties. Based on the resulting predictions of job runtimes a novel scheduling algorithm is able to estimate the latest job start time necessary to meet the requested deadline and sort the queue accordingly to minimise the amount of deadline overrun. The testing of the proposed approach was done using the actual job trace collected from a production Grid facility. The best performing execution time predictor (the auto-regressive moving average method) coupled to workload partitioning based on three simultaneous job properties returned the median absolute percentage error centroid of only 4.75%. This level of prediction accuracy enabled the proposed deadline scheduling method to reduce the average deadline overrun time ten-fold compared to the benchmark batch scheduler. Overall, the thesis demonstrates that deadline scheduling of computational jobs on the Grid is achievable using statistical forecasting of job execution times based on historical information. The proposed approach is easily implementable, substantially self-managing and better matched to the human workflow making it well suited for implementation in the utility Grids of the future.