Temporal Incursions

Temporal Paradoxes, also known as Predestination Paradoxes, can boggle the mind. As Star Trek Voyager's Tuvok put it, they are "usually implausible, although not illiogical." Wikipedia has a great article summarizing these events in many works of fiction. Oddly, the recent film Deja Vu is missing.

From the Wikipedia article, here are some favorites:

In Michael Crichton's novel Timeline, several graduate students who are excavating several medieval castles and towns from 14th century France are given the opportunity to travel back in time to the very place and time period they are studying. On a mission to rescue their Professor, who had left his time machine and gotten lost, the students end up causing some of the historical events they had studied.
Movies in the Terminator series deal with predestination paradoxes. In the first movie, Kyle Reese, the soldier sent back in time to protect Sarah Connor, the future mother of his commander John Connor, ends up fathering John Connor with her. Paralleling this, the Terminator cyborg sent back to kill Sarah is destroyed, but its components are salvaged to form the basis of the artificially intelligent computer network Skynet that will, in the future, send it back in time on its murderous mission. Although this second predestination paradox was established in the movie, the characters managed to destroy the components and prevent it. However, in the third movie, the Terminator reveals that the rise of Skynet is inevitable, and that the events of the second movie only postponed it. He turns out to be correct, as the movie ends with Skynet coming online.

In The Time Machine, the protagonist Alexander Hartdegen invents a time machine to go back in time to stop the death of the woman he loves, Emma. However, whenever he saves her, she dies in another way. He travels to the future to find out why and eventually meet the Übermorlock, a powerful psychic. He tells Alexander that the reason he can't save Emma is because her death is what caused him to make the time machine in the first place. Because without her death, there can be no time machine, he therefore can't go back in time to save her.

In The Butterfly Effect, when his teacher asks him to draw what he wants to be when he grows up, seven-year-old Evan Treborn (Logan Lerman) draws a murderer standing over two corpses with a bloody knife. He quickly "blacks out" the memory of having drawn the picture and never sees the drawing afterwards. In response to the picture, Evan's mother takes him to a doctor, who suggests he write about the incident in a journal. As an adult, Evan (Ashton Kutcher) uses the journal to return to the past and draw the picture. Also, seven-year-old Evan visits his father Jason in a psychiatric hospital and "blacks out" the few minutes that lead Jason to attack Evan until a guard accidentally deals a lethal blow to Jason's head. Later, in order to speak to his dead father, Evan uses his journals to relive the visit, during which he provokes the attack.

In the Star Trek: Voyager episode “Relativity,” Captain Braxton of the future timeship USS Relativity recruits Seven of Nine to prevent the USS Voyager from being blown up by a temporal intruder. Her first two attempts are unsuccessful, and she ends up recruiting Captain Kathryn Janeway to find the intruder who planted the bomb. The intruder turns out to be a future version of Braxton, seeking revenge against Janeway, whom he blames for interfering with the timeline on numerous occasions and causing him to endure a 30-year exile on 20th century Earth (as seen in the episode “Future’s End”). Relativity’s First Officer, Lieutenant Ducane, arrests the present-day Braxton for “crimes he will commit,” and promises Janeway that he will clean up the timeline. How this is to be done, however, or whether the events of the episode will continue to exist if he does so, is never explained. Here, the paradox was called the Pogo paradox (after the phrase “We have met the enemy and he is us” from the Pogo comic strip).