Modern endgame theory
Modern endgame theory refers to the recent developments in endgame theory, which rely on miai counting for calculating the count of a position and value of a move, so as to decide in which order moves should be played.
Modern endgame theory simply allows application of the principle of usually playing in order of decreasing move values. 
Sakauchi Jun Ei has been credited with developing the basic theory of miai counting and as such of modern endgame theory. Subsequently Bill Spight and Robert Jasiek have very much extended the theory and helped making it readily applicable by go players. Mathematicians including Elwyn Berlekamp have developed those parts of the theory close to combinational game theory and thermography.
Books on modern endgame theory include
- Endgame 2 Values by Robert Jasiek
- Endgame 3 - Accurate Local Evaluation by Robert Jasiek
- Endgame Problems 1 by Robert Jasiek
- Mathematical Go Endgames by Elwyn Berlekamp and David Wolfe
- Rational endgame by Antti Törmänen
- Yose - Absolute Counting? by O Meien
- Dieter Verhofstadt / Practical Endgame — How to apply this theory in practice
- Miai values list — Gives the miai values of a large number of standard positions
 RobertJasiek: PJT asks: "By whom?" I think criticism comes from various people having studied modern endgame theory more than traditional endgame theory offers as a whole, which is little. They realise the shortcomings, inconsistencies and causes for mistakes in traditional endgame theory.
 RobertJasiek: The insufficiency and inconsistency of traditional endgame theory motivates carelessness resulting in very many accidental calculation mistakes and causes evaluation mistakes of the type guessing a type (gote versus sente) or guessing the moments of interrupting local play. Unless a shape is trivial, about every third guess is wrong. Wrong guesses almost always result in wrong evaluation (unless we have a rare case of accidental, correct values at the end of a wrong calculation). Less frequent systematic types of mistakes can also occur (such as not being aware of effectively the non-existence of local double sente). Wrong values cause wrong decision-making.
 RobertJasiek: PJT asks for clarification but this principle is very clear. Ok, as soon as you are aware of its assumption of considering separate local endgames. Then playing in order of decreasing move values is like always first playing the most valuable move (the move with the largest move value). This is usually correct but there are exceptions, of which you know at least one: last valuable move decisions if some local endgame has a follow-up. If you still do not understand the principle, create your sample position with three simple gote endgames, in which either 1, 2 or 3 stones can be connected. You connect 3 before 2 before 1 stones - in the decreasing order of these same move values. (Yes, there could be much more contents on this page, as of 2019-06-03.)
robertjasiek: Modern endgame theory uses miai values. The theory has two flavours: ignoring vs. including infinitesimals. Usually, Bill and I ignore them in the theory of our flavour and so your second question. CGT is mostly only for the latest endgame.