The main result of this thesis is an original proof that every word has finite width in a compact p-adic analytic group. The proof we give here is an alternative to Andrei Jaikin-Zapirain's recent proof of the same result, and utilises entirely group-theoretical ideas. We accomplish this by reducing the problem to a proof that every word has finite width in a profinite group which is virtually a polycyclic pro-p group. To obtain this latter result we first establish that such a group can be...