This thesis will be a mathematical description of the physical phenomenon entanglement, where the subjects separability and entanglement measures with the main focus being finite dimensional Hilbert spaces, especially the spaces C²⊗C² and C²⊗C³.

The subjects: Tensor products and density matrices are introduced to create a foundation for the rest of the thesis. Following these introductions we define the term entanglement.

We then describe a number of criteria which can be used to determine whether a state is separable or entangled. In the special cases of states on C²⊗C² and C²⊗C³, it is shown that if the state has PPT then the state i separable

Afterwards we take a look at criteria that describe how entanglement measures can be constructed, both through practical usability, and through a axiomatic approach. It is shown that the measures entanglement distillation and entanglement cost, can be used as bound for entanglement measures.

Finally the results will be used in relation to physical examples.