Educators usually mean different constructs when they speak of open tasks: some may refer to pure-mathematics investigative tasks while others may have authentic real-life tasks in mind; some may think of the answer being open while others may refer to an open method. On the other hand, some educators use different terms, e.g. open and open-ended, to mean the same construct, while others distinguish between these terms. It is difficult to hold a meaningful discussion or to define clearly an area of research on open tasks if the idea of what constitutes the construct of openness is vague. Moreover, what students learn depends on the types of tasks that they are given, and different kinds of tasks place differing cognitive demands on students. Thus, the objectives of this article are to clarify the types of mathematical tasks and develop a framework to characterise their openness based on five task variables: goal, method, task complexity, answer and extension; and to discuss how different types of tasks and openness may affect student learning. The openness framework can help teachers to design or select more appropriate tasks to cater to students with different abilities in order to develop in them various kinds of mathematical thinking processes, and it can also make it easier for researchers to study the interaction between different types of openness and student learning.