Organic thin film transistor (OTFT) based device modeling and circuit application is a rapidly emerging research area. Taking cognizance of this fact, our paper reviews various basic to advanced OTFT structures, their performance parameters, materials of individual OTFT layers, their molecular structures, OTFT charge transport phenomena, and fabrication techniques. The performance of p- and n-type conducting polymer and small molecule organic semiconductors are reviewed primarily in terms of field effect mobility, current on/off ratio, and operating voltage for various OTFT structures. Moreover, different organic/inorganic materials for realizing the dielectric layer, electrodes, and the substrate in an OTFT are analyzed. Some of the compact models that are essential for predicting and optimizing the device performance are described that takes into account the mobility enhancement factor and channel length modulation. A detailed study of the single gate, dual gate, vertical channel, and cylindrical gate OTFT structures is carried out. Furthermore, the paper discusses some of the interesting and upcoming applications of organic transistors such as inverters, light emitting diodes (LEDs), RFID tags, and DNA sensors. Although organic transistors boast of a bright future with a wide spectrum of applications, but they still face several challenges in terms of mobility, voltage swings, noise margins, sub-threshold slope, stability, etc., that needs to be resolved to make them a commercially sustainable and viable technology.