Q:

What are some recent developments in density functional theory?

In the book Materials Modelling Using Density Functional Theory: Properties and Predictions by Feliciano Giustino, a timeline of milestones in DFT was given for achievements between 1964 and 1996:
$$\small\begin{array}{|c|c|c|} \hline
\textbf{Year} & \textbf{Milestone} & \textbf{Researchers} \\ \hline
1964, 1965 & \text{HK Theorem/KS Formulation} & \text{Kohn, Hohenberg, Sham}  \\  
1972, 1973 & \text{Relativistic DFT} & \text{von Barth/Hedin, Rajapol/Callway} \\  
1980, 1981 & \text{Local Density Approximiation(LDA)} &\text{Ceperley/Alder, Perdew/Zunger} \\
1984 & \text{TDDFT} & \text{Runge, Gross}  \\
1985 & \text{First Principles MD} & \text{Carr, Parrinello}   \\  
1986 & \text{Quasiparticle Corrections} & \text{Hybertsen, Louie} \\  
1987 & \text{Density Functional Perturbation Theory} & \text{Baroni, Giannozzi, Testa}  \\  
1988, 1993 & \text{Toward Chemical Accuracy} & \text{Lee/Yang/Parr (1988), Becke (1993)} \\
1991 & \text{Hubbard Correction} & \text{Anisimov, Zaanen, Andersen} \\
1992, 1996 & \text{Generalized Gradient Approximation} & \text{Perdew/Burke/Ernzerhof} \\
\hline
\end{array}$$
Has there been any milestone after 1996, or not included in the above list? 
Please limit each answer to one milestone!

Since there's now so many answers that it takes a very long time to scroll down to a specific one, I Have created links for the milestones that have already been explained in an answer:

1993 (Becke) Hybrid Functionals
1995 (Casida) TD-DFRT
2004 (Yanai et al.) Range separation
2005 (Bartlett) ab initio DFT
2006 (Grimme) Double Hybrids
2007 (Becke-Johnson) Dispersion correction
2013 (Kim et al.) Density-corrected DFT
2015 (Sun et al.) SCAN functional

A:

2006 (Grimme): Double hybrid functionals
The timeline of milestones you have given, includes a hybrid functional called B3LYP, which mixes a Hartree-Fock exchange functional with a GGA exchange-functional. In 2006, Stefan Grimme introduced what later became known as "double hybrid functionals", which not only mix the Hartree-Fock exchange functional with a GGA exchange-correlation functional, but also a many-body perturbation theory correction:
\begin{equation}
E_{\textrm{xc}}=\left(1-a_\textrm{x}\right)E_\textrm{x}^{\textrm{GGA}} + a_\textrm{x}E_\textrm{x}^{\textrm{HF}} + bE_\textrm{c}^{\textrm{GGA}} + cE_\textrm{c}^{\textrm{MBPT}}.
\tag{1}
\end{equation}
For the MBPT correction, Grimme used MP2 with the Kohn-Sham orbitals and single-excitations neglected. He tried various double hybrids, and the one that he finally recommended was obtained after setting b=1-c, and then using the Becke-88 functional for $E_\textrm{x}^{\textrm{HF}}$ and the LYP functional for $E_\textrm{c}^{\textrm{GGA}}$. He called this new functional B2PLYP.
By 2011 there existed several different double hybrid functionals made by various other groups, and Goerigk and Grimme created an enormous dataset by combining 30 test sets, containing a grand total of 841 relative energies involving 1218 total energies, and compared the performance of 47 functionals (2 LDA, 14 GGA, 3 meta-GGA, 23 Hybrid, 5 Double Hybrid) on this test suite. Double hybrids were by far the most accurate family of functionals, with an estimated average error of only 1.8 kcal/mol:
                            

A:

2015 (Sun et al.): SCAN functional
The SCAN meta-GGA functional is an extension of the popular PBE GGA [1] and the TPSS [2] and revTPSS [3] meta-GGAs, SCAN adheres to all 17 known exact XC constraints and is constructed to be almost exact for the noble gasses and jellium surfaces. Early evidence suggests that SCAN is more accurate than and of comparable efficiency to the aforementioned GGAs for crystal structure prediction [4-5], ab initio thermodynamics [6-7], and computational catalysis [8].
References:
[1] J. P. Perdew, K. Burke, and M. Ernzerhof,Phys. Rev. Lett.77, 3865 (1996).
[2] J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria,Phys. Rev. Lett.91, 146401 (2003).
[3] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin,and J. Sun,Phys. Rev. Lett.103, 026403 (2009).
[4] H. Peng, Z.H. Yang, J.P. Perdew, and J. Sun, Phys. Rev. X 6, 041005 (2016).
[5] J.H. Yang, D.A. Kitchaev, and G. Ceder, Phys. Rev. B 100, 035132 (2019).
[6] E.B. Isaacs and C. Wolverton, Phys. Rev. Mater. 2, 063801 (2018).
[7] Y. Zhang, D.A. Kitchaev, J. Yang, T. Chen, S.T. Dacek, R.A. Sarmiento-Pérez, M.A.L. Marques, H. Peng, G. Ceder, J.P. Perdew, and J. Sun, Npj Comput. Mater. 4, 9 (2018).
[8] G. Sai Gautam and E.A. Carter, Phys. Rev. Mater. 2, 1 (2018).

A:

Dispersion corrected methods (2007/2010)
Lots of answers already, I would say the main ones are covered. However, in the spirit of the question, I don't think anyone has done dispersion corrections yet. So,
There are many levels of Dispersion corrected methods, but I would say the most common is from Grimme et al. in 2010 (Grimme et. al. 2010 paper.)
The energy correction is calculated as (taken from Frank Jensen's textbook)
\begin{equation}
   \Delta E_{\rm disp} = -\sum_{n=6} s_n \sum_{\rm AB} \frac{C_n^{\rm AB}}{R_{\rm AB}^n}f_{\rm damp}(R_{\rm AB})
\end{equation}
Further variations can also account for higher order $R^n$ dependance.
Becke & Johnson have also done work on more physics based corrections, more can be found at a different question, here.

