Science and spirituality are often seen as two incompatible approaches to reality. This chapter is designed to start bridging this gap. We define science as a joint effort of humans to understand the world and to prevent error, using our senses and invented instruments enhancing our senses. This we call experience of the world in its material aspects. Spirituality can be understood as an effort to understand the general principles or structure of the world through inner experience. There are a few requirements for such an epistemological framework to function. One is that consciousness is understood as complementary to its material substrate, the brain, and hence as capable in principle of having its own access to reality. The other requirement is that dogmatism, both on part of science and on part of religions is put aside and spirituality is understood as a hitherto neglected area of investigation that needs to become part of science as a method of inner experience. Some historical efforts – Roger Bacon's system in the middle ages or Franz Brentano's attempt at the beginning of the history of scientific psychology – can serve as examples. Preconditions and open questions are discussed to pave the way for a better understanding.