Donald Knuth lists, in the preface of The Art of Computer Programming Vol 3, the following as the important questions of design and analysis of algorithms[1]:
How are good algorithms discovered?
How can given algorithms and programs be improved?
How can the efficiency of algorithms be analyzed mathematically?
How can a person chose rationally between different algorithms for the same task?
In what senses can algorithms be proved "best possible"?
How does the theory of computing interact with practical considerations?
How can external memories like tapes, drums, or disks be used efficiently with large databases?
↑Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN0-201-89685-0. Preface, pp.v.