Usually the definition of adiabatic process is that there is no heat or matter transfer between the system and surroundings. This definition is not entirely correct.
Let's take a cylinder divided by a piston. There is a gas in one half of it. Now you move the piston very-very fast, so that the gas can fill all the cilinder's volume. You have moved the piston almost instantly, much faster than the speed of gas molecules. Although there was no mass or heat exchange with surroundings this is not an adiabatic process. F.e. as the molecules of gas spread around all the available volume the total energy of the molecules would not change, so the temperature of gas would not change, etc.
The usual formula of adiabatic process is derived assuming that the parameters of the system are changing very slowly compared to the speed of molecules in the system.
In practice when you deal with actual gases this assumption usually holds. Speed of molecules is about kilometers per second, no pistons moves that fast. But there are some heat leaks. And the slower the process, the greater is the influence of the heat leaks.
So, the adiabatic process is a process slow enough (compared to the speed of particles in the system), and fast enough, so that the influence of heat leaks is not significant.
Theoretical physicists usually have to care about the first part of this definition ("isn't it said that the system is isolated?")
Experimental physicists usually have to care only about the second part.
UPDATE to avoid (or add) some confusion.
Looks like there are different definitions of "adiabatic" process. According to one of them the above-described gas expansion IS called adiabatic.
But neither of definitions says the adiabatic process has to be fast. Making the process fast is only an experimental trick required to reduce the influence of heat leaks.