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The cognitive interference channel is a two-user interference channel in which one transmitter is non-causally provided with the message of the other transmitter. This channel model has been extensively studied in the past years and capacity results have been proved for certain classes of channels. This paper presents new inner and outer bounds for the capacity region of the cognitive interference channel, as well as new capacity results. Previously proposed outer bounds are expressed in terms of auxiliary random variables for which no cardinality constraint of their alphabet is known. Consequently, it is not possible to evaluate such outer bounds explicitly for a given channel. The outer bound derived in this work is based on an idea originally devised by Sato for channels without receiver cooperation and results in an outer bound that does not contain auxiliary random variables, thus allowing it to be more easily evaluated. The inner bound presented in this work-which includes rate splitting, superposition coding, a broadcast channel-like binning scheme and Gel'fand Pinsker coding-is the largest known to date and is explicitly shown to include all previously proposed achievable rate regions. The novel inner and outer bounds are shown to coincide in certain cases. In particular, capacity is proved for a class of channels in the so-called “better cognitive decoding” regime, which includes the regimes in which capacity was known. Finally, the capacity region of the semi-deterministic cognitive interference channel, in which the signal at the cognitive receiver is an arbitrary deterministic function of the channel inputs, is established.