Classic serotonergic hallucinogens or psychedelics produce an altered states of consciousness (ASC) that is characterized by profound alterations in sensory perception, mood, thought including the perception of reality, and the sense of self. Over the past years, there has been considerable progress in the search for invariant and common features of psychedelic states. In the first part of this review, we outline contemporary approaches to characterize the structure of ASCs by means of three primary etiology-independent dimensions including oceanic boundlessness, anxious ego-dissolution, and visionary restructuralization as well as by 11 lower-order factors, all of which can be reliably measured by the altered state of consciousness questionnaire (APZ-OAV). The second part sheds light on the dynamic nature of psychedelic experiences. Frequently, psychedelic subjects progress through different stages over time and levels of changes along a perception-hallucination continuum of increasing arousal and ego-dissolution. We then review in detail the acute effects of psychedelics on sensory perception, emotion, cognition, creativity, and time perception along with possible neural mechanisms underlying them. The next part of this review outlines the influence of non-pharmacological factors (predictors) on the acute psychedelic experience, such as demographics, genetics, personality, mood, and setting, and also discusses some long-term effects succeeding the acute experience. The last part presents some recent concepts and models attempting to understand different facets of psychedelic states of consciousness from a neuroscientific perspective.