Using first principles simulations we perform a detailed study of the structural, electronic, and transport properties of monatomic 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. Second, we find that the averaged conductance shows slight parity oscillations with the number n of atoms in the chain. Additionally, the conductance of chains of fixed n oscillates as the end atoms are pulled apart, due to the gradual closing and opening of conductance channels as the chain straightens.