Psilocybin can occasion mystical-type experiences having substantial and sustained personal meaning and spiritual significance (pdf) by R. R. Griffiths &W. A. Richards & U. McCann & R. Jesse

This double-blind study evaluated the acute and longer-term psychological effects of a high dose of psilocybin relative to a comparison compound administered under comfortable, supportive conditions. […]
The participants were hallucinogen-naðve adults reporting regular participation in religious or spiritual activities. Two or three sessions were conducted at 2-month intervals. […]
When administered under supportive conditions, psilocybin occasioned experiences similar to spontaneously occurring mystical experiences.

The study has elicited some glowing praise from people in the field:

The study by Griffiths et al. is noteworthy both for the rigorousness of its design and execution, as well as the clarity of its results. It demonstrates that psilocybin can be safely studied in normal human beings who do not have a history of hallucinogenic drug use. As would be expected, during the psilocybin session participants showed perceptual changes and labile mood. It is striking that majority of the participants reported 2 months later that the psilocybin-induced experience was personally very meaningful and spiritually significant. Indeed, most of them rated the psilocybin-induced experience as one of the top five most important experiences in their life. It is especially notable that participants reported that the drug produced positive changes in attitudes and behaviors well after the sessions, and these self-observations were consistent with ratings by friends and relatives.
— Charles R. Schuster, Department of Psychiatry and Behavioral Neuroscience,
Wayne State University School of Medicine
The article by Griffiths et al. in this issue of Psychopharmacology should make all scientists interested in human psychopharmacology sit up and take notice. It is the first well-designed, placebo-controlled, clinical study in more than four decades to examine the psychological consequences of the effects of the hallucinogenic (psychedelic) agent known as psilocybin. In fact, one would be hard pressed to find a single study of psychedelics from any earlier era that was as well-done or as meaningful. Perhaps more importantly, despite the notion by many people that psychedelics are nothing more than troublesome drugs of abuse, the present study convincingly demonstrates that, when used appropriately, these compounds can produce remarkable, possibly beneficial, effects that certainly deserve further study.
— David E. Nichols, Department of Medicinal Chemistry and Molecular Pharmacology, Purdue University School of Pharmacy and Pharmaceutical Sciences

Sadly, current NIDA director Nora Volkow felt the need to almost apologetically explain NIDA’s involvement in the research funding.

As the nation’s preeminent drug abuse research organization, NIDA’s mission is to support research and provide information on the addictive and adverse health consequences of drugs of abuse. [emphasis added]

She went on to say that the researchers had kind of gone off on their own.
Yeah, sometimes science happens despite the government’s best efforts.
More reading on this:

Jacob Sullum at Hit and Run
Washington Post
Wall Street Journal
L.A. Times
ABC News
Multidisciplinary Association for Psychedelic Studies (see also The Good Friday Experiment)

Before you rush out and get your own mystical experience, remember a few things:

Psilocybin is currently illegal (unless you’re in the Netherlands).
A supportive, safe environment is essential for a positive experience with hallucinogenic drugs. Attempting things like driving is dangerous and stupid.
Psilocybin can intensify current psychological states. Taking it while undergoing emotional problems can be extremely discomfiting. Same thing is true for those with a history (or family history) of psychological problems or schizophrenia.

Other than those points, Psilocybin is not addictive, and has generally mild physical effects, while the psychoative effects can be quite intense. More at Erowid.