Enough preface already!
The Five Problems of Today's Physics
largely taken from Lee Smolin, The Trouble with Physics
The problem of quantum gravity: Combine general relativity and quantum theory into a single theory, preferably with no infinities, singularities, or universe-eating monsters.
The foundational problems of quantum mechanics: Resolve the philosophical problems in the foundations of quantum mechanics, either by making sense of the theory or by inventing a new theory that does make sense.
The problem of 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 problem of the adjustable constants: Explain how the values of the free constants in the standard model of particle physics are chosen in nature. [I think this one is not such an issue, because I believe that God could have set the constants to whatever He pleased. However, it does seem possible that He might instead have chosen a scheme for nature that sets its own constants; we can see from the universe and from His Word that He loves beauty and order.]
The problem of cosmology: Describe what the universe used to be like and how it got to what it is like now. [Smolin has this as “Explain the dark matter and dark energy.” I have adjusted it to reflect the need to offer a biblically-consistent alternative to the big bang model.]