Abstract: The Arbitrary Order Calculus (AOC) also known as Fractional Calculus (FC) has become as important as the Integral Order calculus (IOC).Studies involving this topic have been used in many fields within Sciences and Engineering.For this work, a historical survey was made upon it and its origin, which is a result of the personal correspondence exchange between Leibniz and Guillaume L'Hôspital and therefore it has been used until now.The main purpose of this work is to present the great contributions of researchers and experts who were crucial to its development, what has made it possible and kept in constant improvement until the present days.On the other hand, it also revealed that along with these contributions, some controversies were detected.After the historical facts have been exposed, the main definitions and properties necessary to the comprehension of the Fractional Calculus and its effective use by Scientists and Engineers were fully presented and explained here.