  Hamiltonian formalisms provide powerful tools for the computation of
approximate analytic solutions of the Einstein field equations. The
post-Newtonian computations of the explicit analytic dynamics and motion of
compact binaries are discussed within the most often applied
Arnowitt-Deser-Misner formalism. The obtention of autonomous Hamiltonians is
achieved by the transition to Routhians. Order reduction of higher derivative
Hamiltonians results in standard Hamiltonians. Tetrad representation of general
relativity is introduced for the tackling of compact binaries with spinning
components. Compact objects are modeled by use of Dirac delta functions and
their derivatives. Consistency is achieved through transition to
$d$-dimensional space and application of dimensional regularization. At the
fourth post-Newtonian level, tail contributions to the binding energy show up
for the first time. The conservative dynamics of binary systems finds explicit
presentation and discussion through the fifth post-Newtonian order for spinless
masses. For masses with spin Hamiltonians are known through
(next-to)$^3$-leading-order spin-orbit and spin-spin couplings as well as
through next-to-leading order cubic and quartic in spin interactions. Parts of
those are given explicitly. Tidal-interaction Hamiltonians are considered
through (next-to)$^2$-leading post-Newtonian order. The radiation reaction
dynamics is presented explicitly through the third-and-half post-Newtonian
order for spinless objects, and, for spinning bodies, to leading-order in the
spin-orbit and spin1-spin2 couplings. The most important historical issues get
pointed out.
