The hardest problem of theoretical physics is the quantum nature of gravity (QG). The leading candidate theory is string theory. String theory attempts to give a complete unified theory of everything, ie. all interactions for all matter. At the moment string theory faces several problems like having an infinite number of equally acceptable vacuum states. Each vacuum represents, no more no less, a universe. This situation has lead many prominent physicists to accept the anthropic principle: the universe is as it is because we are here to observe it. The hope of string theorists is to find a way to determine the "correct" vacuum somehow, or at least to give probabilities for the different vacua.
Experimental observations requiring new theory include dark energy and dark matter in the universe. These questions are discussed briefly eg. here.
People doing research on various aspects of QG (strings, branes; spin networks, etc) are too numerous to list here. Some, having an own web site, are
Edward Witten of Princeton IAS, String Theory (review)
Joe Polchinski of UC Santa Barbara, The Big Book of String Theory in two vols.
String Cosmology, people All about strings
Leonard Susskind of Stanford, String Theory and the Anthropic Principle
Andrei Linde of Stanford, Inflatory Cosmologies
Paul Steinhartd of Princeton, Cyclic Cosmologies
Lee Smolin of Perimeter Inst., Quantum Gravity
Abhay Ashtekar of Penn State, Mathematics for new physics
Gerard 't Hooft of Utrecht, Everything
Stephen Hawking of Cambridge Black Hole radiation, Wave Function of the Universe
Other important developments include Alain Connes' non-commutative geometry, the holograhic principle, described excellently by Jacob Bekenstein in Scientific American article (August, 2003). The anthropic principle seems to have both defenders and, mostly, strong opponents.
Links to some 'randomly' chosen recent papers (my blog).