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A source of random message bits is to be embedded into a covertext modeled as a discrete memoryless source (DMS), resulting in a stegotext from which the embedded bits should be recoverable. A causal code for such a scenario consists of an encoder that generates the stegotext as a causal function of the message bits and the covertext, and a decoder that reproduces the message bits as a causal function of the stegotext. A semicausal code, on the other hand, has an encoder that is causal only with respect to the covertext, and not necessarily with respect to the message, and has a possibly noncausal decoder. We analyze the possible tradeoffs among: a) the distortion between the stegotext and the covertext, b) the compressibility of the stegotext, and c) the rate at which random bits are embedded, that are achievable with causal and semicausal codes, with and without attacks on the stegotext. We also study causal and semicausal codes for the private version of the above scenario in which the decoder has access to the covertext. Connections are made with the causal rate-distortion function of Neuhoff and Gilbert, as well as the problem of channel coding with causal side information at the transmitter analyzed by Shannon.