The problem of quantum gravity: Combine general relativity and quantum theory into a single theory that can claim to be the complete theory of nature.
The foundational problems of quantum mechanics: Resolve the problems in the foundations of quantum mechanics, either by making sense of the theory as it stands or by inventing a new theory that does make sense.
The unification of particles and forces: Determine whether or not the various particles and forces can be unified in a theory that explains them all as manifestations of a single, fundamental entity.
The tuning problem: Explain how the values of the free constants in the standard model of particle physics are chosen in nature.
The problem of cosmological mysteries: Explain dark matter and dark energy. Or, if they don't exist, determine how and why gravity is modified on large scales. More generally, explain why the constants of the standard model of cosmology, including the dark energy, have the values they do.

Problem 1: The Problem of Quantum Gravity

Quantum gravity is the effort in theoretical physics to create a theory that includes both general relativity and the standard model of particle physics. Currently, these two theories describe different scales of nature and attempts to explore the scale where they overlap yield results that don't quite make sense, like the force of gravity (or curvature of spacetime) becoming infinite. (After all, physicists never see real infinities in nature, nor do they want to!)

Problem 2: The Foundational Problems of Quantum Mechanics

One issue with understanding quantum physics is what the underlying physical mechanism involved is. There are many interpretations in quantum physics - the classic Copenhagen interpretation, Hugh Everette II's controversial Many Worlds Interpretation, and even more controversial ones such as the Participatory Anthropic Principle. The question that comes up in these interpretations revolves around what actually causes the collapse of the quantum wavefunction. (The puzzle of the curious aspect of human consciousness's role in resolving these questions is related in Quantum Enigma.)
Most modern physicists who work with quantum field theory no longer consider these questions of interpretation to be relevant. The principle of decoherence is, to many, the explanation - interaction with the environment causes the quantum collapse. Even more significantly, physicists are able to solve the equations, perform experiments, and practice physics without resolving the questions of what exactly is happening at a fundamental level, and so most physicists don't want to get near these bizarre questions with a 20 foot pole.

Problem 3: The Unification of Particles and Forces

There are four fundamental forces of physics, and the standard model of particle physics includes only three of them (electromagnetism, strong nuclear force, and weak nuclear force). Gravity is left out of the standard model. Trying to create one theory which unifies these four forces into a unified field theory is a major goal of theoretical physics.
Since the standard model of particle physics is a quantum field theory, then any unification will have to include gravity as a quantum field theory, which means that solving problem 3 is connected with the solving of problem 1.

In addition, the standard model of particle physics shows a lot of different particles - 18 fundamental particles in all. Many physicists believe that a fundamental theory of nature should have some method of unifying these particles, so they are described in more fundamental terms. For example, string theory, the most well-defined of these approaches, predicts that all particles are different vibrational modes of fundamental filaments of energy, or strings.

Problem 4: The Tuning Problem

A theoretical physics model is a mathematical framework that, in order to make predictions, requires that certain parameters are set. In the standard model of particle physics, the parameters are represented by the 18 particles predicted by the theory, meaning that the parameters are measured by observation.
Some physicists, however, believe that fundamental physical principles of the theory should determine these parameters, independent of measurement. This motivated much of the enthusiasm for a unified field theory in the past and sparked Einstein's famous question "Did God have any choice when he created the universe?" Do the properties of the universe inherently set the form of the universe, because these properties just won't work if the form is different?

The answer to this seems to be leaning strongly toward the idea that there is not only one universe that could be created, but that there are a wide range of fundamental theories (or different variants of the same theory, based on different physical parameters, original energy states, and so on) and our universe is just one of these possible universes.

In this case, the question becomes why our universe has properties that seem to be so finely tuned to allow for the existence of life. This question is called the fine-tuning problem and has promoted some physicists to turn to the anthropic principle of explanation, which dictates that our universe has the properties it does because if it had different properties, we wouldn't be here to ask the question. (A major thrust of Smolin's book is the criticism of this viewpoint as an explanation of the properties.)

Problem 5: The Problem of Cosmological Mysteries

The universe still has a number of mysteries, but the ones that most vex physicists are dark matter and dark energy. This type of matter and energy is detected by its gravitational influences, but can't be observed directly, so physicists are still trying to figure out what they are. Still, some physicists have proposed alternative explanations for these gravitational influences, which do not require new forms of matter and energy, but these alternatives are unpopular to most physicists.