The obstacles and difficulties that science teachers face when teaching the nature of science (NOS) are well-known. Nonetheless, little is known of what science teacher educators (STEs) know, do, and think about this issue. Thus, a study with 142 Spanish STEs was conducted. This was aimed at analysing (i) what preparation the STEs have to teach NOS, (ii) what educational experience they have about NOS, (iii) what importance they give to NOS in their training plans, (iv) when NOS should be taught, (v) how to integrate NOS in science education, (vi) how to teach NOS, (vii) what to teach about NOS, and (viii) the influences interconnecting the STEs' preparation, experience, and opinions regarding NOS teaching. Among the results, it was found that most STEs state that they have ample knowledge of NOS, are well prepared pedagogically to teach it, and have extensive experience in teaching it. However, there was some mismatch between their stated preparation and their experience teaching NOS. In addition, they think that NOS should be taught from early ages and be treated as cross-cutting content in science education. They also consider that, when teaching NOS, an implicit approach is just as effective as an explicit-reflective one. As for their selection of NOS content to include, they are fundamentally divided into two groups -- those who select only epistemic aspects of NOS and those who select a balanced proportion of both epistemic and non-epistemic aspects. The limitations of the study are reported along with the issues that require further research.