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Within eukaryotes, DNA replication is controlled within the context of the cell cycle. As the cell grows and divides, it progresses through stages in the cell cycle; DNA replication takes place during the S phase (synthesis phase).
In this image DNA is blue, cell structure (microtubules) is green and DNA connection sites with the cell structure (kinetochore) are red. Now as the other answers say, DNA replication takes place at the interphase (which is divided to G1, S and G2 phases) and more specifically at the S phase.