A bubble is defined as a period of substantial price deviations from a fundamental value. Bubble detection and crash prediction are by default quite tough processes. Financial bubbles are of great importance; their growth attracts media attention, even small households are interested on taking part on this fest. In that state of euphoria where a “perpetual money machine” operates, no one could think that things might go worst. In the early literature developed in the 80’s and 90’s there was not clear evidence on detecting bubbles; empirical evidence was poor and there was absence of well developed econometric tools. This research aims to investigate deeper the structure of the bubbles with putting some emphasis on the behavioral perspective as also providing a review to the existing literature. Additionally, in this project a basic bubble detection technique in historical prices and real time data will be implemented. The approaches followed are alternations of the Augmented Dickey Fuller tests; the Supremum ADF, Rolling ADF, Generalized ADF and Backward ADF tests combined with a date-stamping strategy. The historical prices of the Price-Dividend ratio, the Cyclically Adjusted Price-Earnings ratio and the Federal Reserve Overnight Repurchase Agreement Index have been used. The tests try to detect explosive behavior in the US stock exchange markets. This project also aims to identify any explosiveness in the current US stock exchange market, implying the entry in a new bubble territory.