When the interaction strength in a many-body system  becomes  larger than the electronic bandwidth, one  enters the uncharted territory of extremely correlated quantum systems. Familiar methods such as perturbation theory, are of limited value here. On the other hand   experimental realizations of such quantum materials, including the not-so-new cuprates, have yielded detailed  an often surprising body of 

data,  over a large array of  accessible properties. This has created a serious gap in our understanding the basic physics of these systems.

The  extremely correlated Fermi liquid theory of the (strongly correlated)  tJ model was formulated in 2011 to circumvent the above mentioned large coupling problem.  In this colloquium I will introduce  the basic qualitative ideas behind this analytical  theory,  the resulting equations at low orders, and some results from it.

I will present calculations   in 1-2- and infinite dimensions highlighting the resistivity and ARPES spectra, and  test these against  exact numerical results from DMRG (1-d) and DMFT (infinite dimensions).  In the  2 dimensional case relevant to cuprates, one finds  an astonishingly small effective Fermi  temperature scale from the resistivity.  I will also present a few predictions   from the theory.