All the various notions of oriented theories. Example of oriented theories. Are there oriented theories which are not representable?
Most theories are oriented, but Witt groups are not! I have notes on an example which apparently proves this, by saying that any oriented theory will have connecting homomorphisms equal to zero in the setting of a closed subscheme with open complement (affine?????). Baptiste Calmes computes what happens for Witt groups.
Oriented Borel-Moore homology, Oriented homology, Orientable cohomology, Oriented cohomology
Motivic cobordism, Algebraic cobordism
See Levine for more examples.
nLab page on D50 More about oriented theories