In grid architecture, a grid workflow system is a type of high-level grid middleware which aims to support large-scale sophisticated scientific or business processes in a variety of complex e-science or e-business applications such as climate modelling, disaster recovery, medical surgery, high energy physics, international stock market modelling and so on. Such sophisticated processes often contain hundreds of thousands of computation or data intensive activities and take a long time to complete. In reality, they are normally time constrained. Correspondingly, temporal constraints are enforced when they are modelled or redesigned as grid workflow specifications at build-time. The main types of temporal constraints include upper bound, lower bound and fixed-time. Then, temporal verification would be conducted so that we can identify any temporal violations and handle them in time. Conventional temporal verification research and practice have presented some basic concepts and approaches. However, they have not paid sufficient attention to overall temporal verification effectiveness and efficiency. In the context of grid economy, any resources for executing grid workflows must be paid. Therefore, more resources should be mainly used for execution of grid workflow itself rather than for temporal verification. Poor temporal verification effectiveness or efficiency would cause more resources diverted to temporal verification. Hence, temporal verification effectiveness and efficiency become a prominent issue and deserve an in-depth investigation. This thesis systematically investigates the limitations of conventional temporal verification in terms of temporal verification effectiveness and efficiency. The detailed analysis of temporal verification effectiveness and efficiency is conducted for each step of a temporal verification cycle. There are four steps in total: Step 1 - defining temporal consistency; Step 2 - assigning temporal constraints; Step 3 - selecting appropriate checkpoints; and Step 4 - verifying temporal constraints. Based on the investigation and analysis, we propose some new concepts and develop a set of innovative methods and algorithms towards more effective and efficient temporal verification. Comparisons, quantitative evaluations and/or mathematical proofs are also presented at each step of the temporal verification cycle. These demonstrate that our new concepts, innovative methods and algorithms can significantly improve overall temporal verification effectiveness and efficiency. Specifically, in Step 1, we analyse the limitations of two temporal consistency states which are defined by conventional verification work. After, we propose four new states towards better temporal verification effectiveness. In Step 2, we analyse the necessity of a number of temporal constraints in terms of temporal verification effectiveness. Then we design a novel algorithm for assigning a series of finegrained temporal constraints within a few user-set coarse-grained ones. In Step 3, we discuss the problem of existing representative checkpoint selection strategies in terms of temporal verification effectiveness and efficiency. The problem is that they often ignore some necessary checkpoints and/or select some unnecessary ones. To solve this problem, we develop an innovative strategy and corresponding algorithms which only select sufficient and necessary checkpoints. In Step 4, we investigate a phenomenon which is ignored by existing temporal verification work, i.e. temporal dependency. Temporal dependency means temporal constraints are often dependent on each other in terms of their verification. We analyse its impact on overall temporal verification effectiveness and efficiency. Based on this, we develop some novel temporal verification algorithms which can significantly improve overall temporal verification effectiveness and efficiency. Finally, we present an extension to our research about handling temporal verification results since these verification results are based on our four new temporal consistency states. The major contributions of this research are that we have provided a set of new concepts, innovative methods and algorithms for temporal verification in grid workflow systems. With these, we can significantly improve overall temporal verification effectiveness and efficiency. This would eventually improve the overall performance and usability of grid workflow systems because temporal verification can be viewed as a service or function of grid workflow systems. Consequently, by deploying the new concepts, innovative methods and algorithms, grid workflow systems would be able to better support large-scale sophisticated scientific and business processes in complex e-science and e-business applications in the context of grid economy.