Answering this depends on the meaning of 'promising':

1) 'promising' as most likely to achieve the high levels of functionality, sophistication of design, intelligence, etc. Don't think that was intended, so will ignore.

2) 'promising' as most likely to impact and alter society.

3) 'promising' as offering the most benefit to society.

and maybe some other meanings I've missed.

Looking at [3] - benefit to society, I suggest this is areas of risk where robots can replace humans, such as emergency rescue, firefighting, etc - where robots can reduce risk to humans or save humans from situations a rescue worker couldn't.

Looking at [2] - most likely to impact and alter society, I suggest two zones - health care and education, and warfare. Robots in health care and education replace humans in caring roles and remove the social externalities of caring. Once this really starts to become widespread, we will see extreme reactions against it, including many on a religious basis. For example, educational and health care robots are literally evil under Christian Realism because they lack a spirit. Christian Realism is the most popular form of Christian philosophy in US politics (eg: Obama). From a military perspective, the prospect of killing people without risk to your own personnel is so attractive that it is inevitable robots will be widely deployed in warfare, and may completely replace troops. Afterall, how can you argue for risking your citizen's lives if they can be replaced by a machine? Much of military strategy is focused on supplying troops and limiting your own casualities. We have no idea how robot armies will alter that thinking, but I expect it will lead to much greater willingness to go to war. From a business perspective, this is great - huge market in which the customer destroys the product and needs more. If I had a robot soldier company, I would be lobbying for laws which forbid human soliders.

How "robot" is defined will clarify this question, but I would say that robots and robotics are going to further impact "general human activity" and "interpersonal communication." Right now, smart home technologies and robotics are advancing. Google mini/Alexa, Roomba, Next, etc.

I had thought similarly to Sousan Arafeh: we need to split the analysis across the different robotic solutions including:

The landscape in each of the above is evolving super fast so it is not easy to predict which is the most promising. In my area of practice I am already immersed in (1) but have already had clients asking about (4).