Guessing from a piece of information is what humans do in their reasoning processes, that is why reasoning and obtaining conjectures can be considered almost equivalent. These reasoning processes are nothing else than posing new questions which possible answers are non contradictory with the available information. This is the idea that allows to introduce different mathematical models by means of different conjecture operators, built up depending on how the concept of non-contradiction is understood. A relevant contribution of this chapter is that there can be conjecture operators not coming from Tarski's operators of consequences, in this way untying the concept of conjecture from a previously given deductive system. The concept of a conjecture proves to include those of logical consequences, hypotheses and speculations.