Radiotherapy delivered using scanned beams of protons enables greater conformity between the dose distribution and the tumour than conventional radiotherapy using X rays. However, the dose distributions are more sensitive to changes in patient anatomy, and tend to deteriorate in the presence of motion. Online dose calculation during treatment delivery offers a way of monitoring the delivered dose in real time, and could be used as a basis for mitigating the effects of motion. The aim of this work has therefore been to investigate how the computational power offered by graphics processing units can be harnessed to enable fast analytical dose calculation for online monitoring in proton therapy. The first part of the work consisted of a systematic investigation of various approaches to implementing the most computationally expensive step of the pencil beam algorithm to run on graphics processing units. As a result, it was demonstrated how the kernel superposition operation, or convolution with a spatially varying kernel, can be efficiently implemented using a novel scatter-based approach. For the intended application, this outperformed the conventional gather-based approach suggested in the literature, permitting faster pencil beam dose calculation and potential speedups of related algorithms in other fields. In the second part, a parallelised proton therapy dose calculation engine employing the scatter-based kernel superposition implementation was developed. Such a dose calculation engine, running all of the principal steps of the pencil beam algorithm on a graphics processing unit, had not previously been presented in the literature. The accuracy of the calculation in the high- and medium-dose regions matched that of a clinical treatment planning system whilst the calculation was an order of magnitude faster than previously reported. Importantly, the calculation times were short, both compared to the dead time available during treatment delivery and to the typical motion period, making the implementation suitable for online calculation. In the final part, the beam model of the dose calculation engine was extended to account for the low-dose halo caused by particles travelling at large angles with the beam, making the algorithm comparable to those in current clinical use. By reusing the workflow of the initial calculation but employing a lower resolution for the halo calculation, it was demonstrated how the improved beam model could be included without prohibitively prolonging the calculation time. Since the implementation was based on a widely used algorithm, it was further predicted that by careful tuning, the dose calculation engine would be able to reproduce the dose from a general beamline with sufficient accuracy. Based on the presented results, it was concluded that, by using a single graphics processing unit, dose calculation using the pencil beam algorithm could be made sufficiently fast for online dose monitoring, whilst maintaining the accuracy of current clinical systems.