A brief review of the best studied mindfulness/mediation program (Mindfulness Based Stress Reduction) is available via the unscientific but still useful Wikipedia, as well as from Psychology Today. You can also learn more about Mindfulness Based Stress Reduction Program  directly from the University of Massachusetts Medical School’s webpage; the program was created by professor emeritus Jon Kabat-Zinn, and the Center for Mindfulness still operates MBSR clinics.

Additionally, a very brief and non-exhaustive list of some of the many studies on meditation and mindfulness can be found below. As these are all individual scientific studies or reviews, they will not be the best resource for those without training in evaluation of scientific methods specific to the respective fields conducting the research (including clinical psychology and medicine). Instead, the point of providing scientific literature on the topic is 1. to provide further reading for those with some understand of these fields, and 2. supporting my claims that meditation is indeed and evidence based practice (as well as to provide descriptions of what types of meditative practices have been studied). 

Evans S, Ferrando S, Findler M, Stowell C, Smart C, Haglin D. Mindfulness-based cognitive therapy for generalized anxiety disorder. J Anxiety Disord. 2008 May;22(4):716–721. 

Hofmann SG, Sawyer AT, Witt AA, Oh D. The effect of mindfulness-based therapy on anxiety and depression: A meta-analytic review. J Consult Clin Psychol. 2010 Apr;78(2):169–183. 

Hoge, E.A. et al., (2013) Randomized Controlled Trial of Mindfulness Meditation for Generalized Anxiety Disorder: Effects on Anxiety and Stress Reactivity. J Clin Psychiatry. 74(8): 786–792.  doi: 10.4088/JCP.12m08083

Kabat-Zinn J, Massion AO, Kristeller J, et al. Effectiveness of a meditation-based stress reduction program in the treatment of anxiety disorders. Am J Psychiatry. 1992 Jul;149(7):936–943.

Krygier, J.R., et al., (2013). Mindfulness meditation, well-being, and heart rate variability: A preliminary investigation into the impact of intensive Vipassana meditation. International Journal of Psychophysiology. 89; 305–313

Niazi, A.K, & Niazi, S.K., (2011). Mindfulness-based stress reduction: a non-pharmacological approach for chronic illnesses. N Am J Med Sci. 3(1): 20–23. doi: 10.4297/najms.2011.320

Tacón, A.M., et al., (2003). Mindfulness Meditation, Anxiety Reduction, and Heart Disease: A Pilot Study.  Family & Community Health 26(1); 25-33
van der Zwan, J.E., de Vente, W., Huizink, A.C. et al. Physical Activity, Mindfulness Meditation, or Heart Rate Variability Biofeedback for Stress Reduction: A Randomized Controlled Trial. Appl Psychophysiol Biofeedback 40, 257–268 (2015). https://doi.org/10.1007/s10484-015-9293-x
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Vollestad J, Sivertsen B, Nielsen GH. Mindfulness-based stress reduction for patients with anxiety disorders: evaluation in a randomized controlled trial. Behaviour research and therapy. 2011 Apr;49(4):281–288.