Can Buddhists see into the future? It's said that Buddhist practice can cause the arising of certain psychic powers. This answer hints that one of those physic powers might be precognition. Have I understood that correctly? I've never read anything about this in any of the texts. I wonder if anyone else has and can provide a reference where precognition is written about.
Pa Auk Sayadaw treats this topic in Knowing and Seeing, P. 190-196.

According to the Sayadaw, follow the method given for discerning the past, making appropriate changes to see the future, including future lives. He says there are multiple methods and this is one of them.


  How You Discern the Past
  
  To discern the past, you begin by making an offering of either
  candles, flowers, or incense at a pagoda, or to a Buddha image. You
  should make a wish for the rebirth you desire, for example, to become
  a monk, nun, man, woman, or deva.
    Afterwards, you should go and sit in
  meditation, develop concentration, and discern in turn internal and
  external mentality and materiality. This is neces- sary, because if
  you cannot discern external mentality and materiality, you will have
  great difficulty discern- ing past mentality and materiality. That is
  because the discernment of external mentality and materiality is
  similar to the discernment of past mentality and materiality.  Then you
  should discern the mentality and materiality that occurred at the time
  of making the offering at the pagoda or Buddha image, as if they were
  an external object. When doing this, an image of yourself at the time
  of offering appears. You should discern the four elements in that
  image.  When the image breaks into kalàpas, discern all 190 the types
  of materiality in the six doors, especially the fifty-four types in
  the heart-base. Then you will be able to discern the bhavaïga
  consciousnesses, and the mind- door thought-processes that that arise
  in-between. You should discern those mind-door thought-processes
  backwards and forwards, and find the defilement-round (kilesavañña)
  mind-door thought-process with twenty mental formations, and
  kamma-round (kammavañña) mind-door thought-process
  (manodvàra-vãthi) with thirty- four mental formations.  Let me
  illustrate with a practical example: the case of making an offering of
  candles, flowers, or incense to a Buddha image, and making a wish to
  be reborn to become a monk.  In this case, ignorance is to deludedly
  think that ‘a monk’ is an ultimate reality; craving is the desire and
  longing for life as a monk; and clinging is the attach- ment to life
  as a monk. These three, ignorance, craving, and clinging, are all
  found in the consciousness that makes up the round of defilements
  (kilesa-vañña).  If, instead of making a wish to be reborn to become
  a monk, you had made a wish to be reborn to become a woman, then the
  ignorance would be to deludedly think that ‘a woman’ is an ultimate
  reality; craving would be the desire and longing for life as a woman;
  and clinging the attachment to life as a woman.  In the examples,
  volitional formations (saïkhàra) are the wholesome intentions
  (kusala-cetanà) of the offering, and kamma is their force of kamma.
  Both are found in the consciousnesses that make up the kamma round of
  dependent-origination.  When you are thus able to discern the
  mentality and materiality of the defilement- and kamma-round of the
  recent past, you should go back to the more distant past time previous
  to the offering, and in the same way discern the mentality and
  materiality. Then go back a little further again, and repeat the
  process. In this way, you discern the mentality and materiality of one
  day ago, one week ago, one month ago, one year ago, two years ago,
  three years ago and so on. Eventually you will be able to discern
  right back to the mentality-materiality of the rebirth-linking
  consciousness (pañisandhi-citta) which arose at the conception of
  this life.  By looking for the causes of conception, you go back even
  further, and see either the mentality- materiality of the time near
  death in the previous life, or the object of the near-death
  impulsion-consciousness (maraõàsanna-javana-citta). There are three
  possible objects for the near death impulsion: 
  1. Kamma; again having the thoughts that pro- duced a particular good or bad action in the past, for example, an offering.
  2. Kamma sign (kamma-nimitta); for example, a pagoda, a monk, flowers, or an object offered.
  3. Rebirth sign (gati-nimitta); the place where you will be reborn. For a human rebirth it is the future mother’s womb, and is usually red
  like a red carpet.  If you discern the mentality-materiality near
  death, you will also discern the object of the near death
  impulsion, be it kamma, kamma sign, or rebirth sign. This object
  appears because of the force of kamma which produced the
  rebirth-linking consciousness (pañisandhi-citta). When you discern
  this, you are able to discern also the volitional formations and kamma
  that produced the resultant aggre- gates of this life, and the
  preceding ignorance, craving, and clinging. After that, you should
  discern the other mental formations of that kamma- and
  defilement-round. 
  
  How You Discern the Future
  
  Once the power of this insight-knowledge has been de- veloped by
  discerning the causes and effects through those past lives, you can,
  in the same way, discern the causes and effects in future lives. The
  future you will see, and which may still change, is the result of both
  past and present causes, one of which is the medita- tion you are
  doing. To discern the future, you begin by discerning the present
  materiality-mentality, and then look into the future until the time of
  death in this life. Then either the kamma, kamma sign, or rebirth sign
  will appear, because of the force of a particular kamma you performed
  in this life. You will then be able to discern the rebirth-linking
  mentality-materiality to be produced in the future life.  You must
  discern as many lives into the future as it takes till ignorance
  ceases without remainder. This happens with the attainment of the
  arahant path(arahatta-magga), that is, your own attainment of
  ara- hantship. You should then continue discerning into the future,
  until you see that the five aggregates, mentality- materiality, cease
  without remainder, that is, at the end of the arahant life, at your
  own Parinibbàna. Thus you will have looked into the future, and seen
  the complete cessation of phenomena (dhamma).  Discerning the five
  aggregates of the past, present, and future, and also discerning their
  causal relation, is what I call the fifth method. Having completed the
  fifth method, you can now learn what is called the first method, the
  one taught by the Buddha.  The first method of discerning dependent-
  origination (pañiccasamuppàda) goes over three lives, and in forward
  order. It begins with the causes in the past life, that is, ignorance
  and volitional formations. They cause the results in the present life:
  the rebirth-linking consciousness, mentality-materiality, the six
  sense- bases, contact, and feeling. There are then the causes in this
  life, craving, clinging, and becoming, which cause the results of
  birth, ageing, death, and all forms of suffering in the future life. 
  You have to look for ignorance, craving and clinging in the defilement
  round, see how it causes the kamma round, and how the kammic force of
  the kamma round in turn causes the five aggregates of conception.  That
  concludes my brief explanation of how to discern dependent-origination
  according to the fifth and first methods. There are many more details
  which you can learn by practising with a proper teacher.


 The Buddha makes predictions about the future in the suttas and, in one case, if I recall correctly, takes action to prevent a disciple from having a bad rebirth; it is a female disciple who is a weaver or something if I am correct. If anyone knows what I am referring to, please do leave a comment.