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Goppa codes are special linear codes which are used as keys for the McEliece and Niederreiter cryptosystems. Goppa codes with non-trivial automorphism groups form weak keys for the cryptosystems. However, it has been conjectured that automorphism groups of Goppa codes are trivial in general. This thesis uses subfield subcodes of Generalised Reed-Solomon (GRS) codes to study the structure of Goppa codes. This helps us give a subset of Goppa codes that have non-trivial automorphism groups. We also give a non-identity automorphism which is common to most non-binary Goppa codes.