The notions of strongly consecutive repeat free code and weakly consecutive repeat free code were introduced by Pebody in his paper in the Journal of Combinatorial Theory Series A in 2006. This thesis aims to investigate the the maximum sizes of such codes, in particular in the case when the length is fixed and the alphabet size is large. Pebody constructs a strongly consecutive repeat free code of maximal size, which he calls the alternating code. We show that the size of an alternating code is polynomial in the alphabet size, give methods for computing this polynomial and explicitly determine the most significant coefficients of this polynomial in terms of the sequence of 'up/down numbers' and related sequences. Pebody defines a family of codes (which we call Pebody codes) that are weakly consecutive repeat free codes. Pebody conjectures that for all parameters there exists a member of this family that is a weakly consecutive repeat free code of maximal size. We show that the maximal size of a Pebody code agrees closely with the maximal size of a strongly consecutive repeat free code. We use techniques from combinatorics and functional analysis, together with computational results, to give estimates for the leading terms of the maximal size of a Pebody code of fixed length when the alphabet size is large.