Mathematics has its famous Millennium Problems (and, historically, Hilbert's 23), questions that helped to shape the direction of the field.

I have little idea, though, what the Riemann Hypotheses and P vs. NP's of statistics would be.

So, what are the overarching open questions in statistics?

Edited to add:
As an example of the general spirit (if not quite specificity) of answer I'm looking for, I found a "Hilbert's 23"-inspired lecture by David Donoho at a "Math Challenges of the 21st Century" conference: High-Dimensional Data Analysis: The Curses and Blessings of Dimensionality

So a potential answer could talk about big data and why it's important, the types of statistical challenges high-dimensional data poses, and methods that need to be developed or questions that need to be answered in order to help solve the problem.

A big question should involve key issues of statistical methodology or, because statistics is entirely about applications, it should concern how statistics is used with problems important to society.

This characterization suggests the following should be included in any consideration of big problems:


How best to conduct drug trials.  Currently, classical hypothesis testing requires many formal phases of study.  In later (confirmatory) phases, the economic and ethical issues loom large.  Can we do better?  Do we have to put hundreds or thousands of sick people into control groups and keep them there until the end of a study, for example, or can we find better ways to identify treatments that really work and deliver them to members of the trial (and others) sooner?
Coping with scientific publication bias.  Negative results are published much less simply because they just don't attain a magic p-value.  All branches of science need to find better ways to bring scientifically important, not just statistically significant, results to light.  (The multiple comparisons problem and coping with high-dimensional data are subcategories of this problem.)
Probing the limits of statistical methods and their interfaces with machine learning and machine cognition.  Inevitable advances in computing technology will make true AI accessible in our lifetimes.  How are we going to program artificial brains?  What role might statistical thinking and statistical learning have in creating these advances?  How can statisticians help in thinking about artificial cognition, artificial learning, in exploring their limitations, and making advances?
Developing better ways to analyze geospatial data.  It is often claimed that the majority, or vast majority, of databases contain locational references.  Soon many people and devices will be located in real time with GPS and cell phone technologies.  Statistical methods to analyze and exploit spatial data are really just in their infancy (and seem to be relegated to GIS and spatial software which is typically used by non-statisticians).

