Most missiles do use rocket engines. Solid rockets are much easier to store and use than liquid fueled engines, whether rocket or turbojet. They can get pretty

Most missiles do use rocket engines. Solid rockets are much easier to store and use than liquid fueled engines, whether rocket or turbojet. They can get pretty fast, too: the AMRAAM is supposed to go mach 4. The downside of any rocket engine is that the fuel and oxidizer run out quickly; the AMRAAM goes that fast in order to have a long range, as the motor burns out fairly soon. A few newer air to air missiles, like the British Meteor, have air breathing liquid fueled turbojets in order to increase range. Cruise missiles with their air breathing engines are the odd one out in missile tech.