Understanding the foundations of quantum mechanics, and in particular the measurement problem. This may result in a radical modification of our conception of physical reality.

The development of a quantum mechanically consistent theory of gravity. The reconciliation of general relativity with quantum theory is absolutely essential for challenge (3) below.

The unification of all the fundamental forces in nature. Supersymmetric string theories offer a promising prospect in this regard.

Related to the above is a determination of the nature of dark matter on the scale of clusters and superclusters of galaxies.

On a different front, an understanding of the origin of life, and a firm determination of the existence or non-existence of extraterrestrial intelligent life.

Again on a different front, an understanding of the operation of the human brain at the fundamental level.