The fundamental limits of communication over state-dependent discrete memoryless channels with noiseless feedback are studied, under the assumption that the communicating parties are allowed to use variable-length coding schemes. Various cases are analyzed, with the employed coding schemes having either bounded or unbounded codeword lengths, and with state information revealed to the encoder and/or decoder in a strictly causal, causal, or non-causal manner. In each of these settings, necessary and sufficient conditions for positivity of the zero-error capacity are obtained and it is shown that, whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. Moreover, it is shown that the vanishing-error capacity of state-dependent channels is not increased by the use of feedback and variable-length coding. Both these kinds of capacities of state-dependent channels with feedback are thus fully characterized.