The invention provides container closure assemblies, especially for carbonated beverage containers, having substantially helical second thread segment (20) with a profiled lower thread surface (24). The profiled lower thread surface (24) includes a first, lower region (28) with a low pitch and a second region (30) with a steep pitch. The thread (20) may also include a step feature (32) to block unscrewing of the closure at an intermediate position where gas venting can take place, and a third region (34) of the said lower surface having a low pitch adjacent to the step. The closures are easy to apply and remove, and do not blow off unpredictably or uncontrollably under pressure.