ACCELERATION IN RELATIVITY AND QUANTISATION OF GRAVITY
Mashhoon and others are attempting to go beyond present theory. I do not hold out much hope for their success. It is difficult to understand their motivation and I am not clear that it addresses the problems I raise. A brief discussion is available in portable document format.
APPLICATIONS OF CLIFFORD ALGEBRAS IN PHYSICS
Also known as geometric algebras. Formulation of the minimal extension of Maxwell’s theory to curved spacetime and indepth investigation of Chisholm and Farwell’s spin gauge theories which predicted the top quark mass without a Higgs mechanism. A good reference regarding general applications of these algebras across theoretical physics is:

C. Doran, A. Lasenby: Geometric Algebra for Physicists, Cambridge University Press (2003)
References for the fascinating spin gauge theories are:

J.S.R. Chisholm, R.S. Farwell: Electroweak spin gauge theories and the frame field, J. Phys. A Math. Gen. 20, 65616580 (1987)

J.S.R. Chisholm, R.S. Farwell: Unified spin gauge theory of electroweak and gravitational interactions, J. Phys. A Math. Gen. 22, 10591071 (1989)

J.S.R. Chisholm, R.S. Farwell: Unified spin gauge theories of the four fundamental forces, Proc. of the IMA Conf.: The Interface of Mathematics and Physics, Oxford University Press, Oxford (1990)

J.S.R. Chisholm, R.S. Farwell: Tetrahedral structure of idempotents of the Clifford algebra C(3,1). In: Clifford Algebras and their Applications in Mathematical Physics, ed. by A. Micali et al., Kluwer Academic Publishers (1992) pp. 2732

J.S.R. Chisholm, R.S. Farwell: Unified spin gauge theories of the four fundamental forces. In: Clifford Algebras and their Applications in Mathematical Physics, ed. by A. Micali et al., Kluwer Academic Publishers (1992) pp. 363370

J.S.R. Chisholm, R.S. Farwell: A fermionboson mass relation and the top mass, J. Phys. G Nucl. Part. Phys. 18, L117L122 (1992)
I have detailed notes on this but not all in electronic form. One of the most interesting things about this theory is that one can obtain the usual relations for the W and Z masses without recourse to the Higgs mechanism. It just shows that what one can do with mathematics is almost limitless!
COSMOLOGICAL REDSHIFT
Demonstrations of redshift in standard RobertsonWalker cosmologies directly from the minimal extension of Maxwell’s equations (MEME) to those spacetimes, by finding exact solutions to those equations (in several different ways, including conformal transformation of solutions in flat spacetime). No references here. Anyone could do this. Compare with the usual derivation of redshift which makes no mention of the underlying electromagnetic theory MEME!
FREE FALL
Motion of ‘freely’ falling spinning bodies in curved spacetimes to demonstrate that the axis of rotation will be approximately FermiWalker transported along the worldline and the worldline will not be exactly a geodesic. This is analysed in an appendix of the following reference, but I have detailed notes:

B.S. DeWitt: Bryce DeWitt’s Lectures on Gravitation, ed. by S.M. Christensen, Springer, Heidelberg (2011)
WHEELERFEYNMAN THEORY OF ELECTROMAGNETIC RADIATION
Currently under investigation by Dürr et al. in Munich. The original references are:

J.A. Wheeler, R.P. Feynman: Interaction with the absorber as the mechanism of radiation, Rev. Mod. Phys. 17, 157 (1945).

J.A. Wheeler, R.P. Feynman: Classical electrodynamics in terms of direct interparticle action, Rev. Mod. Phys. 21, 425 (1949)
I have detailed notes on the first of these two papers and raise two criticisms which seem to me to throw doubt on implementation of this idea. Recent work in Munich can be found in:

G. Bauer, D.A. Deckert, D. Dürr: MaxwellLorentz dynamics of rigid charges, arXiv:1009.3105v2 [mathph] 15 Apr 2011

G. Bauer, D.A. Deckert, D. Dürr: On the existence of dynamics of WheelerFeynman electromagnetism, arXiv:1009.3103v2 [mathph] 9 Aug 2011
These are mathematical results applying fixed point theorems. It remains to be seen whether the original idea can be implemented physically. A recent paper addressing this issue is

G. Bauer, D.A. Deckert, D. Dürr, G. Hinrichs: On irreversibility and radiation in classical electrodynamics of point particles, J. Stat. Phys. 154, 610622 (2014); arXiv:1306.3756 [physics.classph] 17 Jun 2013
My comments on this paper can be downloaded as a pdf here.
I am not keen on this theory, although it does seem there is something that needs to be explained regarding the way the radiation reaction can be made to 'drop out' if there is some physical substance to the equations. The reason for this position is my personal expectation that all 'particles' will always turn out to be made of other things, i.e., they will never occupy only a point in space. But we know how to get the radiation reaction when the charged particle is actually a spatial distribution of charge. Furthermore, we know that bound states of charged particles have some 'electromagnetic mass', and we lose our explanation for this in the WheelerFeynman picture.