What is the general hierachy and structure of academic research and how do they differ around the world?

My (very limited) experience with academic research is that in North America the structure goes something like this: Professor applies for grants, some projects receive funding, those projects are then headed by PhD students and undergrad helpers with regular meetings with the prof to ensure the group is making progress. How does this differ in, say, Europe and Asia? Are the professors more involved with the groups or does it really just differ on a prof to prof basis?