Shotokan karate is a hobby of mine. I say 'hobby' but for me it's become so much more than that. It's not just that I find myself practising my age-uke (pronounced 'aggey-uki') or various favourite kata whenever I have some space free around me; it's more in relation to how it's been a route for me to live a lot more healthier [active] lifestyle more generally. I am indeed at one with myself...
A recent paper by Sandra Origua Rios and colleagues  provides some good evidence that practising the martial arts - the 'hard martial arts such as karate, taekwondo or kung fu - carries quite a few health benefits for adults. Based on their searching the collected peer-reviewed research literature and use of the Effective Public Health Practice Project tool, authors concluded that most studies on such training carried positive effects "showing some improvement and maintenance of balance, cognitive function and psychological health" when practised by adults. Studies like this one from Witte and colleagues  tell you quite a lot about what you need to know. All this is good news indeed to the many, many practitioners of such martial arts. Also an added boost to the masses that will undoubtedly take up one or more of the arts when they watch the very best demonstrate their skills at the 2020 Olympics...
Of course it's not necessarily all 'positive' when it comes to the use of martial arts and health and wellbeing. The authors note that 'risk of injury' is always going to be a concern when engaging in disciplines that rely on strikes, blocks and throws and 'confrontation of force by force'. In the context that risk of injury is a risk carried by most, if not all sporting activities, the authors add that the majority of injuries sustained in the hard martial arts tend to be minor; particularly when practising at less than elite competitive level (as most people do). Indeed, on the specific point of injury risk, I can add a few things: (a) developing 'control' when sparring (we call it 'kumite') is a key point that comes from practice and more practice, and (b) protective equipment such as mitts and on occasion, gum shields and various, ahem, parts-of-body protectors, are available and sometimes used. Other injuries such as bruising, blistering and some occasional joint pain/stiffness are typically short-lived and clear up after a few days of rest. I might also mention that actually facing an opponent and 'not fearing getting hit' is something else important that comes from such training, given that most people aren't used to being in such an atypical situation and tend to panic (and that's when injuries can happen).
Origua Rios et al also suggest that more research is required on the risk-benefit profile of the hard martial arts including evidence based on randomised controlled trials (RCTs) tracking "morbidity and mortality in the long term." I would agree; but with the proviso that looking at hard martial arts vs. no martial arts training is not necessarily going to be the only methodological way forward in light of other, more general evidence talking about sedentary behaviour and health-related quality of life for example. Indeed, when one talks about 'cognitive functions' potentially being positively affected by martial arts training, it would be useful to see comparisons with the more 'soft' martial arts such as Tai Chi for example, to see how and why such as effect might originate and perhaps generalise. I have my ideas about where any maintenance/improvement to cognitive abilities may originate from in karate terms (try learning and retaining knowledge of all those kata!) but more scientifically-derived information is required.
I know that the hard martial arts probably won't be everyone's cup of tea. I know some people might think it's a bit too rough or challenging or something for those with sprightly athleticism who can 'do a really good high kick'. I would however caution against such preconceptions. As my instructor(s) often say: quality of movement is better than speed or height of said movement (or words to that effect).
And with physical activity (PA) and cognitive functions in mind, it's perhaps also timely that a recent meta-analysis from Celia Álvarez-Bueno and colleagues  has also been published talking about how: "PA, especially physical education, improves classroom behaviors and benefits several aspects of academic achievement, especially mathematics-related skills, reading, and composite scores in youth." I daresay that getting more children and young people involved in the martial arts might have a similar effect; although ensuring that due care and attention is paid to 'enjoyment' as being a big part of why someone does or does not stick to a particular sport (see here).
 Origua Rios S. et al. Health benefits of hard martial arts in adults: a systematic review. J Sports Sci. 2017 Nov 21:1-9.
 Witte K. et al. Comparing the effectiveness of karate and fitness training on cognitive functioning in older adults—A randomized controlled trial. Journal of Sport and Health Science. 2016; 5: 484-490.
 Álvarez-Bueno C. et al. Academic Achievement and Physical Activity: A Meta-analysis. Pediatrics. 2017. Nov 22.