Subtle but well characterized structural phase transformations occur at low temperatures in some lanthanum cuprate superconductors. These transformations, which involve collective tilting of the CuO6 octahedra, have a strong influence on superconducting and normal state properties. Experiments suggest that for a given hole doping the following hierarchy exists for superconductivity in these materials: Tc(HTT)>Tc(LTO)>Tc(LTT), where HTT represents the untilted structure and LTO and LTT are the two limiting low temperature tilted structures. An additional very strong suppression of Tc, suggestive of a competing electronic groundstate, occurs in the LTT phase for a hole to Cu ratio of 1/8. Neither of these effects are understood at present. This paper reviews both the evidence upon which these conclusions are based and some outstanding unanswered questions.