Using first principles simulations we perform a detailed study of the structural, electronic and transport properties of monoatomic platinum chains, sandwiched between platinum electrodes. First we demonstrate that the most stable atomic configuration corresponds to a zigzag arrangement that gradually straightens as the chains are stretched. Secondly, we find that the conductance at equilibrium atomic spacing does not oscillate with the number of atoms n in the chain, but instead decreases almost monotonically with n. In contrast, the conductances of chains of fixed n oscillate as the end atoms are pulled apart, due to the gradual closing and opening of conductance channels as the chain straightens.