The serotonergic classical psychedelics include compounds that primarily activate the brain's serotonin 2 A receptor (5-HT2AR), such as LSD, psilocybin, and DMT (ayahuasca). The acute effects of these compounds are well-known as are their ability to increase the emotional state both in healthy people and in those with neuropsychiatric disorders. In particular psilocybin, the psychoactive constituent in "magic mushrooms", has shown great potential for treatment of anxiety and depression. A unique and compelling feature of psychedelics is that intake of just a single psychedelic dose is associated with long-lasting effects. This includes effects on personality, e.g., higher openness, and amelioration of depressive symptoms. This review focuses on these stunning effects and summarizes our current knowledge on which behavioral, biochemical, neuroimaging, and electrophysiological data support that the intriguing effects of psychedelics on the human brain and mind are based on neural plasticity. The review also points to so far understudied areas and suggests research questions to be addressed in future studies which potentially can help to understand the intriguing long-term effects after intake of a single (or a few) psychedelic doses.