According to the big bang theory the universe is approximately 13.7 billion years old. The oldest stars in our galaxy are also around this age.
"Within the given uncertainties, this number [for the Milky Way] also fits very well with the current estimate of the age of the Universe, 13,700 million years, that is the time elapsed since the Big Bang. It thus appears that the first generation of stars in the Milky Way galaxy was formed at about the time the “Dark Ages” ended, now believed to be some 200 million years after the Big Bang."
Therefore, the milky way is about 13.5 billion years old. I attached a link for a more detailed process on aging the milky way.
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