Organic field-effect transistors (OFETs) usually operate in the accumulation mode, where the biasing of the gate induces the charging of the insulator-semiconductor interface; the bias is negative in the most common case of p-type semiconductors. We show here that the application of a positive gate bias leads to the formation of a depletion layer, and a subsequent decrease of the drain current. We develop an analytical model for this depletion mode of the OFET. It is shown that measurements in the depletion mode give access to parameters such as the doping level and the density of trap levels. The model is applied to data obtained on sexithiophene (6T) film and single crystal OFETs. A substantial amount of traps is found in unsubstituted 6T, whereas the dihexyl substituted 6T is practically trap free. 6T single crystals are characterized by a very low doping level, which can be related to their very high purity. The possible use of the depletion mode to increase the on-off current ratio of OFETs is discussed.