This thesis studies surfaces with fibrations by curves of small genus, usually locally in a neighbourhood of a singular fibre. The main method is algebraic, consisting of describing the relative canonical algebra. In the genus 2 case there are classical results of Horikawa that have been used successfully by Xiao Gang to get global results on surfaces. This thesis is mainly concerned with the genus 3 case, which is much harder; even here the results are not definitive. We prove that for genus 2 and 3 the relative canonical algebra is generated in degrees 1, 2, 3 and related in degree ≤ 6. In fact we give a much more detailed analysis of the ring by generators and relations.