  This paper reviews progress on the Fractional Quantum Hall Effect (FQHE)
based on what we term hamiltonian theories, i.e., theories that proceed from
the microscopic electronic hamiltonian to the final solution via a sequence of
transformations and approximations, either in the hamiltonian or path integral
approach, as compared to theories based on exact diagonalization or trial
wavefunctions. We focus on the Chern-Simons (CS) approach in which electrons
are converted to CS fermions or bosons that carry along flux tubes and our
Extended Hamiltonian Theory (EHT) in which electrons are paired with
pseudo-vortices to form composite fermions (CF) whose properties are a lot
closer to the ultimate low-energy quasiparticles. We address a variety of
qualitative and quantitative questions: In what sense do electrons really bind
to vortices? What is the internal structure of the Composite Fermion and what
does it mean? What exactly is the dipole picture? How are (approximate)
transport gaps, zero-temperature magnetic transitions, the temperature
dependent polarizations of gapped and gapless states, the NMR relaxation rate
$1/T_1$ in gapless states, and gaps in inhomogeneous states computed? It is
seen that though the CS and EHT approaches agree whenever a comparison is
possible, results that are transparent in one approach are typically opaque in
the other, making them truly complementary.
