In my talk I will discuss the stimulating aspects of conjectures in modern mathematics. Examples of questions and conjectures in number theory and algebraic geometry will be discussed in order to illustrate the concepts mentioned. I will present and propose what criteria could be given for a question to be called a conjecture. Most material will be presented in a way accessible for a general mathematical audience. Although part of my lecture is "about mathematics" there will be enough mathematical contents to satisfy those of you who also want to see proofs and facts, beautiful structures, expectations, and problems to work on.