[horsebox]

I know that a liquid will boil when its vapour pressure becomes greater than the atmospheric pressure but what happens if the liquid is trapped in a container with no air at all? Would it just cease to boil? Would the temperature rise well above its boiling point?

[Schrödinger]

Isn't that the case where critical behaviour comes into play? That's when the vapour phase and liquid phase cease to exist as 2 different phases. Rather, they become one phase.. a dense fluid.

[horsebox]

First time I've heard of this. Is pressurised gas considered to be in both phases?

[Borek]

Over critical point there is no gas nor two phases. There is one phase, that is neither liquid nor gas.

[rabolisk]

The temperature would rise above the normal boiling point, in equilibrium with its vapor. There is always some vapor present above the liquid in a container. As the temperature rises, the equilibrium shifts (always) in favor of the gas, (and pretty dramatically so) but the liquid does not completely convert to gas at the normal boiling point. At normal boiling point, there is enough vapor in the container to exert 1 atm of pressure. At higher temperature, there is more. Above the critical temperature, the entire substance is in one uniform phase called the supercritical fluid.