Clean water and/or sewage. Almost a billion people don't have access to either, because the infrastructure doesn't exist for municipal sewage or water treatment. Huge numbers of people die from preventable diseases, simply because their water is fouled.

Until cellphones, most of the world didn't have reliable communications, because nobody could build the necessary infrastructure. Wireless changed that. We need the equivalent breakthrough for water treatment.

Dean Kamen is doing interesting stuff in this area (a stirling-engine based system for water distillation [1]), but his approach is still limited by costs and distribution.

Solving this problem would literally change the world, and unlike many of the suggestions here, it's an area where there's hope that a small research team could make a dent (for example, Kamen wanted to raise $1M for Slingshot.)

[1] https://en.wikipedia.org/wiki/Slingshot_(water_vapor_distill...