I am a neuroscience researcher studying meditation and mindfulness, and also a longtime practitioner of Tibetan Buddhism. Experience suggests to me that it's hard enough for any scientist to remain unbiased about her/his own work, and that can only be more challenging for a researcher studying something that he has already received enormous personal benefits from—like myself and meditation. So, I have chosen to cultivate an attitude of healthy skepticism about my own field.
In light of that, I'll echo Rabbit, whose answer states that the scientific field is young and it's too early to have formed any reliable scientific conclusions. I partially agree that the methodology of most of the studies is "flawed"; however, I think it's more fair to say that the methodology of early exploratory studies was necessarily limited, but now that widespread interest has developed higher quality studies are being designed. This is just an inevitable growth process of a field; high-quality human research studies are extremely expensive and that kind of funding is simply not available until after a certain amount of preliminary evidence has accumulated.
In order to understand the scientific literature on this topic, I think it's important to understand a few distinctions.
One is the distinction between Buddhist meditation and practice as a system designed to reduce suffering and cultivate compassion by softening the mental habit of grasping at a solid sense of self; and "mindfulness" as a broadly used term for a recently fashionable trend in psychological therapy which had its origins primarily in the aforementioned Buddhist tradition, but which has taken on a secular life of its own. These two categories, secular mindfulness vs. Buddhist practice, are frequently poorly distinguished in the scientific literature, and in general usage of key terms like "mindfulness" and "meditation" is somewhat sloppy overall. Some publications reporting "benefits of mindfulness" are actually looking at comparisons of groups of, say, long-time Tibetan Buddhist practitioners vs. controls, which is a very different thing than, say, inexperienced novices randomly assigned to a MBSR group for eight weeks, which is in turn very different from an individual-differences study where randomly selected people's scores on a mindfulness psychological questionnaire are shown to correlate with some other values of interest.
Another important thing to understand is the distinction between specific and non-specific effects. This is a big issue in psychology in general; for example, Bruce Wampold has a large body of evidence showing that psychotherapy in general mainly works because of the "therapeutic alliance" between the therapist and patient, and the specific type of therapy is virtually irrelevant. Likewise, our lab has published research showing no differences between MBSR and another, non-mindfulness-based health and well-being intervention on a range of general measures (see MacCoon et. al. 2012 and 2014 at that link). This is NOT to say that the interventions in question are "ineffective"; to the contrary, they are quite effective. What is missing is scientific evidence regarding how, specifically, one unique technique is effective, and how it might be uniquely different in its effects from the wide range of various things one might do to improve physical and mental health. This is kind of a double-edged sword: on the one hand, it's good news, because it means the science supports the idea that you don't have to worry too much about doing exactly the right thing. On the other hand, it's bad news, because in some cases it might be important to choose exactly the right thing, and we don't know how to do that yet.
Now I'm going to say a few things about what I think might be hinted at but not yet proven in the scientific literature, with regard to specific benefits related to Buddhist practice and/or the aspects of mindfulness that are most closely related thereto.
As I mentioned briefly above, the more or less uniquely Buddhist practice of suffering-reduction is based on reducing the habit of grasping at a sense of solid and enduring self that is threatened by change and uncertainty. There are a few areas where I think (but certainly can't prove yet) that the scientific literature is on the right track for this.
One starts with Jon Kabat-Zinn's own early seminal publications on MBSR. The first paper published on the topic states, "the SR&RP [earlier name for MBSR] functions as a 'net' to catch patients who tend to 'fall through the cracks' in the health care delivery system, neither improving in their medical system over time nor feeling satisfied with the results of the traditional medical management of their problem(s)... Many chronic pain patients ultimately receive the verdict that 'you're going to have to learn to live with this.' The SR&RP helps teach patients the how of living with chronic pain." The implication here is that the early clinical successes reported for MBSR were in populations that were receiving it as a last resort. They were highly motivated because all other promises of relief had failed, and they were highly primed by their last-resort status to accept the message of acceptance and non-striving that is part of mindfulness. In other words, the motivation was there to truly let go of something deep about themselves. I would say that this process is necessary for any truly specific effects of mindfulness; and the level of motivation necessary makes it very unlikely that this process will show up in studies with random assignment to groups, or non-last-resort populations. In summary, I think the literature hints that mindfulness has unique and specific benefits for populations who are truly at a last resort and have no choice but to accept acceptance.
The other regards autoimmune conditions. Among the early MBSR research was also a finding of benefit for psoriasis. Also, the work of Melissa Rosenkranz in our lab has been steadily closing in on a specific function of mindfulness for fine-tuning the immune system. Note that Dr. Rosenkranz's papers at that link do not yet fully make that case, but I am hopeful that her work is moving toward putting the pieces together to show that mindfulness, even relative to the active control intervention, has unique benefits in training the immune system to properly distinguish biological "self" from infection, which is the necessary condition for improving autoimmune conditions. This is particularly intriguing given the fundamental role of the psychological "self" in Buddhist practice and mindfulness.
So, in conclusion, I'd say that there is a lot of evidence of non-specific benefits of Mindfulness, Buddhist practice, etc. There is no conclusive evidence yet of any effects specific to any of the different meanings of the word "mindfulness". However, there is some evidence beginning to accumulate for some specific beneficial effects, particularly in areas regarding appropriately high levels of motivation, and conditions that are relevant to the uniquely self-related aspects of Buddhism/mindfulness.
Now, all that aside, my own personal experience of benefit from my Buddhist practice is a very different story from all that, but that's not really the topic of this question!