Aerodynamic problems can be classified in a number of ways. The first classification criterion is whether the flow is external or internal. External aerodynamics is the study of flow around solid objects of various shapes. For instance, evaluating the lift and drag on an airplane, the shock waves that form in front of the nose of a rocket, and the flow of air over a hard drive head are examples of external aerodynamics. Internal aerodynamics is the study of flow through passages in solid objects. For instance, internal aerodynamics encompasses the study of the airflow through a jet engine or through an air conditioning pipe.

A second classification of aerodynamic problems is the ratio of the problem characteristic flow speed to the speed of sound. A problem is called subsonic if all the speeds in the problem are less than the speed of sound, transonic if speeds both below and above the speed of sound are present (normally when the characteristic speed is approximately the speed of sound), supersonic when the characteristic flow speed is greater than the speed of sound, and hypersonic when the flow speed is much greater than the speed of sound. Aerodynamicians disagree over the precise definition of hypersonic flow; minimum Mach numbers for hypersonic flow range from 3 to 12. Most aerodynamicians use numbers between 5 and 8.

A third way to classify aerodynamic problems is by the importance of viscosity in the flow. In some problems, in which viscous effects on the solution are negligible, viscosity can be safely ignored and set to zero. The approximations to these problems are called inviscid flows. Flows for which viscosity cannot be neglected are called viscous flows.

One of the major goals of aerodynamics is to predict the aerodynamic forces on aircraft.

The four forces that act on a powered aircraft are lift, weight, thrust, and drag. Weight is the force due to gravity and thrust is the force generated by the engine. Lift and drag are aerodynamic forces. Lift is defined as the aerodynamic force acting perpendicular to the direction of travel of the aircraft relative to the surrounding air, and drag is defined as the aerodynamic force acting parallel to the direction of travel. Lift is positive upwards and drag is positive rearwards.

Gases are composed of molecules which collide with one another and solid objects. In aerodynamics, however, gases are considered to have continuous quantities. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at infinitely small points, and are assumed to vary continuously from one point to another. The discrete, molecular nature of a gas is ignored.

The continuity assumption becomes less valid as a gas becomes more rarified. In these cases, statistical mechanics is a more valid method of solving the problem than aerodynamics.

- Conservation of mass: Matter is not created or destroyed. If a certain mass of fluid enters a volume, it must either exit the volume or increase the mass inside the volume.
- Conservation of momentum: Also called Newton's second law of motion
- Conservation of energy: Energy is neither created nor destroyed. It may only change from one manifestation to another.

Note that these laws are based on Newtonian Mechanics, they are not applicable in Einsteinian Mechanics.

Transonic problems are arguably the most difficult to solve. Flows behave very differently at subsonic and supersonic speeds, therefore a problem involving both types is more complex than one in which the flow is either purely subsonic or purely supersonic.

Supersonic aerodynamic problems are those involving flow speeds greater than the speed of sound. Calculating the lift on the Concorde is an example of a supersonic aerodynamic problem.

Supersonic flow behaves very differently from subsonic flow. The speed of sound can be considered the fastest speed that "information" can travel in the flow. Gas travelling at subsonic speed diverts around a body before striking it, it can be said to "know" that the body is there. Air cannot divert around a body when it is travelling at supersonic speeds. It continues to travel in a straight line until it reaches a shock wave and decelerates to subsonic speeds. Mathematically expressed, supersonic flow is hyperbolic while subsonic flow is elliptic.

Another example of the difference between supersonic and subsonic flow is the behaviour in a convergent duct (known as a nozzle in subsonic flow and a diffuser in supersonic flow). Subsonic flow in a convergent duct accelerates and supersonic flow decelerates.

See also: Aeronautics, Fluid dynamics, Bernoulli's equation, Navier-Stokes equations, Center of pressure