Compton scattering from protons and neutrons provides important insight into 
the structure of the nucleon. We present a new extraction of the static 
electric and magnetic dipole polarizabilities of the proton and neutron from 
all published elastic data below 300~MeV in Chiral Effective Field Theory. 
Including the $\Delta(1232)$ as explicit degrees of freedom is particularly 
important for the proton above about $100$~MeV and for deuteron Compton 
scattering at $\gtrsim90$~MeV as measured at SAL and MAXlab. Consistency 
arguments dictate including the $np$-rescattering states and automatically 
render the correct Thomson limit, shedding new light on Weinberg's 
power-counting of nuclear forces. We show that the static electric and 
magnetic scalar polarizabilities of the proton and neutron are identical 
within the accuracy of available data. In view of ongoing efforts at 
HI$\gamma$S, MAMI and MAXlab, we address in detail single- and 
doubly-polarized observables with linearly or circularly polarized 
photons on both un-, vector- and tensor-polarized deuterons. Several 
observables can be used to extract not only spin-independent nucleon 
polarizabilities, but also the so-far practically undetermined spin-dependent 
polarizabilities which parameterize the stiffness of the nucleon spin in 
external electro-magnetic fields. Amongst the questions addressed are 
convergence of the expansion when including the $\Delta$ and the r\^ole 
of $np$-rescattering in few-nucleon systems.