Can anyone explain non-self or Anatta of Buddhism in simple terms with example? Recently I got to know about 'Non-Self' or Anatta term but online material was not comprehendible. Below are my questions:
1) Can anyone explain it in simple terms with example?
2) If there's no permanent self or soul then how did Buddha HIMSELF saw his previous lives in various forms, which would be impossible without a permanent soul. In the case of temporary soul it would have been only limited to current life.
3) What were the cases/scenarios from which Buddha made the 'Non-Self' conclusion?
4) How can I cross verify this claim. I mean experience this (as Buddha said believe on your own experience &amp; then follow &amp; not because I said it.)
1) Can anyone explain it in simple terms with example?


'Self' is a subjective ever-changing 'idea' or 'thought' created by the ignorant mind. For example, the 'self' the mind believed it was at 4 years old is different to the 'self' the mind believes it is at 35 years old. If the mind stops thinking, it can see the body &amp; mind do not have any real self. For example, if an axe is used to cut off a finger, is there any 'self' in that detached finger? Is there any 'self' in an ear lobe or eye-ball or liver or kidney? Is there any 'self' in a feeling of pain, perception of 'green' or sight of object or hearing of a sound? 'Self' is only an idea, such as when a mind believes: "John is a nice person"  but another mind believes: "John is a bad person". 


  2) If there's no permanent self or soul then how did Buddha HIMSELF saw his previous lives in various forms, which would be impossible
  without a permanent soul. In the case of temporary soul it would have
  been only limited to current life.


The Buddha did not see any previous lives. This is a mistranslation &amp; misinterpretation of the original teachings (even though at a later time in Buddhism, scriptures such as Jataka Tales were created that depicted literal past lives). The Buddha recollected 'past abodes' or 'past adherences' ('pubbe nivasa'), which means viewing when in the past when the mind ignorantly attached to either one or all of the five aggregates as "self". This is clearly explained in the Khajjaniya Sutta. Please carefully read the entire Khajjaniya Sutta, very carefully, particularly the 2nd paragraph.


  3) What were the cases/scenarios from which Buddha made the 'Non-Self' conclusion?


The first explanation is the Anatta-lakkhana Sutta. Here, the Buddha started by explaining if each of the five aggregates can be subject to sickness/disease &amp; cannot ultimately be controlled, how can they be a 'self'? Then the Buddha explained how can any of the five aggregates, each being impermanent &amp; unable to bring lasting happiness, be a permanent lasting self or belong to a permanent lasting self? 


  4) How can I cross verify this claim. I mean experience this (as Buddha said believe on your own experience &amp; then follow &amp; not because
  I said it.)


Meditation. Stop thinking and allow the mind to become exclusively observant. Although the sense of self will initially remain, as the mind becomes more pure &amp; naturally 'concentrated', the sense of self, which is merely subtle thoughts, will dissolve. Then the mind (rather than "you") will realise it &amp; everything else is not a self. 

However, be warned: this runs against the nature of instinct. Nature creates the instinct in self in people. Only relatively few minds clearly realise not-self. For most Buddhists, non-self is just blind-faith or a logical deduction (philosophical theory), such as for the famous philosopher Nāgārjuna.  

If needing to read more, this is the best explanation: Anatta &amp; Rebirth by Buddhadasa.