Here is a list of some important questions in biology that interest me:
1) What is the role of protein dynamics in determining protein structure, function, folding, regulation and evolution? Can we use this knowledge to improve how we design new drugs? (see my post
here for more info).
2) To what extent is evolution deterministic? If we were to "replay the tape of life" (i.e. go back in the evolutionary history of an organism and let it evolve again under the same or similar circumstances), would you get the same result? To what extent can we predict how a system will evolve and use that knowledge to fight disease (e.g. pathogens or cancer)?(see my post
here)
3) Only 1% of the human genome codes for protein, and only ~ 10% appears to be evolutionarily conserved. What is the rest of our DNA doing? (see this
thread for more info).
4) How is our DNA organized spatially in the nucleus? What molecular mechanisms determine this organization and how does it impact gene expression and genome evolution? (here's a
review article on the topic).
5) Can we design new organisms from scratch in the lab? What is the "minimal" genome required for life? Is it possible to computationally model an entire organism? (see posts
here and
here)
Of course, not all of these have a good chance of being even partially solved in the next 10-20 years, but there are people willing to take a shot at some of them. Note that I've completely ignored neuroscience (outside my expertise), which many would argue is the frontier for research in biology over the next few decades (or at least the major driver of https://www.whitehouse.gov/share/brain-initiative ).