Abstract. This paper presents a novel proposal to define task parallelism in OpenMP. Task parallelism has been lacking in the OpenMP language for a number of years already. As we show, this makes certain kinds of applications difficult to parallelize, inefficient or both. A subcommittee of the OpenMP language committee, with representatives from a number of organizations, prepared this proposal to give OpenMP a way to handle unstructured parallelism. While defining the proposal we had three design goals: simplicity of use, simplicity of specification and consistency with the rest of OpenMP. Unfortunately, these goals were in conflict many times during our discussions. The paper describes the proposal, some of the problems we faced, the different alternatives, and the rationale for our choices. We show how to use the proposal to parallelize some of the classical examples of task parallelism, like pointer chasing and recursive functions.