Authorities in the U.K. have thwarted three domestic terror plots in "recent months," plots that would have "certainly" been deadly, the head of MI-5 said today.

MI-5 Director General Andrew Parker did not to provide details about the plots in a speech at Thames House, except to say they were foiled with the help of MI-5's partners.

"But we cannot be complacent," he said, according to a copy of his speech provided to ABC News. "Although we and our partners try the utmost, we know that we cannot hope to stop everything."

Parker delivered the speech today a little more than 24 hours after two gunmen stormed into the offices of a French satirical magazine in Paris and murdered several cartoonists. In all, 12 people, including two policemen, were killed in the daytime massacre. As of this report, the gunmen are still on the loose.

"It is too early for us to come to judgments about the precise details or origin of the attack, but it is a terrible reminder of the intentions of those who wish us harm," Parker said.

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Just after Parker's speech began, U.S. officials said the French government believes at least one of the suspects in the attack on the Charlie Hebdo magazine once trained with al Qaeda in the Arabian Peninsula (AQAP), an al Qaeda branch based in Yemen.

But Parker spent much of his speech talking about the terror threat from groups in Syria and Iraq - especially the threat posed by British citizens who travel to the warzone and could return with combat experience and violent plans.

"Around 600 extremists are among the many Britons who have travelled there. A significant proportion has joined ISIL [ISIS]," Parker said, referring to the al Qaeda splinter group that has cut a violent swath through Syria and Iraq.