[CDATAHere I list all theories which do not fit anywhere else, or which I do not understand well enough to fit somewhere.]
Topological chiral homology, probably same as Factorization homology
Algebroid cohomology and Differentiable cohomology
Gelfand-Fuchs cohomology (two spellings)
Good pants homology
Morse homology
Graph homology and graph cohomology
Contact homology, Contact cohomology, Embedded contact homology, Floer homology, Heegard Floer homology, Instanton Floer homology, Kuranishi homology
Khovanov homology, Knot homology, Knot contact homology
Cohomology of generalized homogeneous spaces
L^2 cohomology L^2-de Rham cohomology L^2-Higgs cohomology
Related: Moderate cohomology, Temperate cohomology, and Formal cohomology. See link under Moderate cohomology. Related to D-modules, with applications to Laplace transform.
nLab page on O20 CTs in (non-algebraic) geometry