The benefits of meditation are proven by science.

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Stress:

3,500 adults experience stress reduction (Pub Med)
Even a few minutes works. What about a consistent daily practice? (Forbes)
Eight-week study of mindfulness meditation reduced inflammation caused by stress (Science Direct)
The many benefits of meditation in the classroom (Medical Xpress)
Research has shown that meditation may also improve symptoms of stress-related conditions (Pub Med), including irritable bowel syndrome (Pub Med), stress associated with disease (Journal)post-traumatic stress disorder (Pub Med) and fibromyalgia (Pub Med)
More on how meditation decreases stress (see Psyc Net and  Psychosomatic Medicine)

Aging:

Defeating a key problem of aging (Business Insider).
Seven-year follow-up on people who meditate shows lasting cognitive gains (Science Daily).
A review of 12 studies found that meditation increased attention, memory and mental ability in older individuals (Pub Med).

Mental Performance/Productivity:

Mindfulness meditation improves cognition: Evidence of brief
mental training (Research)
Brain Scans Show How Meditation Improves Mental Focus (Psychology Today)
Improves your ability to multitask (Citation)
Increases your focus & attention (Springer and Plos Biology and Science Direct)
One of the nation’s largest insurance company, Aetna, offer a wellness program where 13,000 employees who have participated in at least one meditation class have reported a 28% reduction in their stress levels, a 20% improvement in sleep quality and said they have gained 62 minutes of productivity per week per person. Aetna

Overall Brain Health:

Increases grey matter (Science Direct) and here (Mindful)
Increases cortical thickness related to attention ( NCBI)
Eight week meditation evaluation (Harvard)

Improves Overall Health/Pain Management:

Increases immune function (Journal) and ScienceDirect)
Decreases pain (NCBI)
In a National Institutes of Health study on meditation: Over a 19 year period 23% decrease in mortality, a 30% decrease in death due to cardiovascular problems, a 49% decrease in cancer mortality as well. “This effect is equivalent to discovering an entirely new class of drugs (but without the inevitable side effects).” American Journal of Cardiology
Decreases inflammation at the cellular level (Science Direct 1 and Science Direct 2 and Science Direct 3)

Anxiety:

Reduced symptoms of a variety of anxiety disorders (Research Gate).
May assist in the control of job-related anxiety. This study examined the reduction of anxiety in a group of nurses (Psyc Net).
2,466 participants also showed meditation may reduce anxiety levels (Wiley).
Mindfulness may ease anxiety (Harvard Medical School)
A three year follow up on individuals who meditate (Science Direct).

Emotional Heath/Happiness:

Teachers who practiced meditation in a short yet intensive program were more calm and compassionate (University of California SF)
Those who mediated vs. those who did not. Those who meditated showed measurable changes in positive thinking and optimism (Journal).
Increases positive emotion (Psychosomatic Medicine and Psyc Net)
Decreases depression (Springer)
Decreases anxiety (Liebert Pub and Psychiatry Online)
Improves social connection & emotional intelligence (Psyc Net)

Improved Self Awareness:

Better state of human well-being. Research showing mindfulness meditation revealing distinct neural modes of self-reference. (Oxford)

Substance Use Disorders/Addictions:

Mindfulness-Based Interventions Modulate Structural Network Strength In Patients With Opioid Dependence (Science Trends)
Mindfulness benefits in cases of substance abuse (Pub Med)
Recovering alcoholics who meditate better controlled their cravings and craving-related stress (NCBI).

PTSD:

Making a Difference: Helping local veterans heal through meditation and yoga (WJAC)

Attention and Communication:

Even meditating for a short period helps. (ScienceDirect).
May even reverse patterns in the brain contributing to a wandering mind, worrying and poor attention (Pub Med).

An increase in Kindness:

This research demonstrated the ability of meditation to increase peoples’ compassion toward themselves and others (Pub Med).
Loving-Kindness and Compassion Meditation: Potential for Psychological Interventions (NCBI)

Improving Sleep:

A comparison of sleep patterns of those who meditate and those who don’t. (Pub Med).
Mindfulness Meditation and Improvement in Sleep Quality (JAMA)