Following is a list of things that need to be dealt with (these are essentially engineering and scientific problems, political problems will need to be dealt with in the shorter term):
Mature nanotechnology: as demonstrated by the RepRap, it is becoming clear that nanotechnology (coupled with another technology, see below) will sort out quite a few of our problems). Once material wealth can be “made” by a machine that itself only requires energy and raw mass (the most advanced post-nanotech replicators will presumably need only energy) then a significant proportion of the iniquities in life will be resolved and done away with.
Fusion: in order to supply a crowded planet with sufficient energy whilst maintaining the integrity of the biosphere for future generations (and for ourselves, see below) it is necessary to create an elegant fusion reactor that produces significantly more energy than it consumes. This will remove any further material iniquity. We will have energy “too cheap to meter”, and the means of production will be owned by anyone and everyone. I suspect this will result in a sort of libertarianism.
Gerontology: the Great Aubrey de Gray believes that if we approach the problem of aging as a purely engineering problem we can overcome it and achieve “engineered negligible senescence”. This is a worthy cause, and will help to remove another great source of iniquity, that which is bestowed by an essentially hostile universe.
Transhumanism: is, as far as I can see, the only contemporary philosophy that has it right with regard to science and technology and adopts the most progressive, mature, and pragmatic attitude towards the potential and the peril of emerging biotechnology and AI. Democratic transhumanism (combining as it does elements of liberal democracy, humanism, and transhumanism) is an ideology that I can believe in and respect. Transhumanism also aims to remove the last of the great iniquities – with material wealth, wealth of resources and so forth dismissed – we would be left with only the basic lack of equality bestowed by our genes and bodies. And then hopefully all the problems would be solved.
The Space Movement: we need to have a (and preferably more than one) functioning self-contained extraterrestrial biosphere, capable of supporting a breeding population of human beings and other animals. My favourite for this one would be an asteroid (e.g. Ceres) hollowed out, a la the Thistledown and inhabited by humans in centrifuges. I hope that some new technology will emerge to allow us to build materials with sufficient tensile strength to overcome the strain of rotating fast enough to produce 9.8 m/s of accelerating centripetal force.
The Ridiculously Huge Telescope: the barriers to understanding currently stand at the limits of the very small and very large. To ensure our survival we need to fully understand our universe. That means we need to see further and in greater detail than before. To do this we need to build a telescope with a baseline as large as the solar system
The Ridiculously Huge Particle Accelerator: we need to see what the fundamental stuff of the universe is. Once we’ve done that we can start really sorting stuff out. To do this we should build a particle accelerator as large as the solar system.