The world witnesses disruption of normal computer functions very often. These disruptions range from personal to business companies to government networks. In this research paper, an attempt has been made to discuss the problems arising out of various forms of cyber crimes, their definition, present laws about cyber crimes and the methodology adopted by cybercriminals. This paper shall also analyze growing use of technology in identity theft, hacking and its use by terrorist organizations. Statistical data is presented to explain how cyber crimes have increased over time and their cost implications. The purpose of this study is also to make the reader aware as to where we stand today in our fight against cybercriminals, what are the future trends and the commitment of various countries of the world for cybersecurity. Since it is being widely accepted that physical devices and human intervention have not proved to be sufficient for protecting cyberinfrastructure, there is a need for more sophisticated cyber defense system which is also robust and flexible. Computing methods using Artificial Intelligence tools have begun to play an important role in cybercrime detection and prevention and their application can be further extended to assist human intelligence in the fight against cyber crimes. This study shall evaluate the progress made till now in the fight against cyber crimes and discuss which Artificial Intelligence tools can be made use of in the fight against this menace.