The Hook Knife is the tool we use to carve concave sections of our spoon, in this case the bowl although it can be used to hollow out other concave areas of the spoon.
The Hook Knife curvature can be simple of compound, you can find them in different geometries.
In the next video we can see Emmet Van Driesche using a Matt White hook knife to hollow an eating spoon bowl:
As you can see, the first cuts are made perpendicular to the grain to be able to hollow without the blade getting stuck in the fibers. When the bowl is fairly defined we can start to make cuts along the grain always following the premises mentioned in: Following the grain: The direction of the cut.
These are some of the Hook Knives available: