The relay channel consists of a transmitter input <i>x</i><sub>1</sub>, a relay input <i>x</i><sub>2</sub>, a relay output <i>y</i><sub>2</sub> , and a receiver output <i>y</i><sub>3</sub>. In this paper, we establish the capacity of three new classes of semi-deterministic relay channels: 1) a class of degraded semi-deterministic relay channels, 2) a class of semi-deterministic orthogonal relay channels, and 3) a class of semi-deterministic relay channels with relay-transmitter feedback. For the first class of relay channels, the output of the relay <i>y</i><sub>2</sub> depends on a deterministic function of the transmitter's input <i>x</i><sub>1</sub>, i.e., on <i>s</i>=<i>f</i><sub>1</sub>(<i>x</i><sub>1</sub>), rather than on <i>x</i><sub>1</sub> directly. In addition, the relay channels satisfy the condition that <i>S</i> → (<i>X</i><sub>2</sub>,<i>Y</i><sub>2</sub>) → <i>Y</i><sub>3</sub> forms a Markov chain for all input probability distributions <i>p</i>(<i>x</i><sub>1</sub>,<i>x</i><sub>2</sub>). Hence, the first class of relay channels includes, but is strictly not limited to, the class of degraded relay channels previously considered by Cover and El Gamal. The partial decode-and-forward strategy achieves the capacity of the class of degraded semi-deterministic relay channels. Next, we consider the class of semi-deterministic orthogonal relay channels where there are orthogonal channels from the relay to the receiver and from the transmitter to the receiver. In addition, the output of the relay <i>y</i><sub>2</sub> is a deterministic function of <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub> and <i>y</i><sub>3</sub> , i.e., <i>y</i><sub>2</sub>=<i>f</i><sub>4</sub>(<i>x</i><sub>1</sub>,<i>x</i><sub>2</sub>,<i>y</i><sub>3</sub>). The class of semi-deterministic orthogonal relay channels is a generalization of the class of deterministic relay channels considered by Kim. The compress-and-forward strategy achieves the capacity of the class of semi-deterministic orthogonal relay channels. For the third class of relay channels, there is a causal and noiseless feedback from the relay to the transmitter. In addition, similar to the second class of relay channels, the output of the relay <i>y</i><sub>2</sub> is a deterministic function of <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, and <i>y</i><sub>3</sub> . Both the generalized strategy of Gabbai and Bross and the hash-and-forward strategy of Kim achieve the capacity of the class of semi-deterministic relay channels with relay-transmitter feedback.
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