Crew scheduling problems have been well studied in a variety of areas within transportation and logistics; however, the application of mathematical modelling techniques within a specifically maritime setting has received relatively little attention in the literature. This research focusses on the crew scheduling problem faced by a large, global company providing support services in the offshore oil industry.Starting with an introduction to crew scheduling and the maritime industry, this thesis goes on to review literature in related areas before giving details of the broader business context and the specific crew scheduling problem in which we are interested. With the company currently using a manual method to update their crew schedules weekly on a rolling basis, often under time pressure, we argue that there is scope for a decision support tool to be introduced which will help the crew planners to find feasible, and potentially good quality, schedules.Two main formulations are presented - a Task-Based model, which makes simplifying assumptions about crew contracts and working patterns, and a more realistic but more complex Time-Windows model. Both of these models can be likened to the crew recovery problems seen in other transportation scheduling literature, and can be solved with the objective of minimizing either the number of changes from the existing schedule or the cost of these changes.While the Task-Based model proved relatively easy to solve, a number of solution methods had to be considered for the Time-Windows problem. Ultimately, a heuristic method was proposed to underpin the scheduling tool. This heuristic method finds an initial solution with a low number of changes, before performing a neighbourhood search which seeks to reduce the solution cost. Results however show there is still much room for improvement, and the thesis concludes with ideas for further extending this research.