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I read different books and anyone has a different explanation.  
Some of them say:

> It does not have a gramamar role and native speakers only know that
> something about that is going to be said after it. The real grammar role is
> done by the 0 pronoun.
>
> It is the subject and sometimes it can become the object.
>
> It is what you are going to comment about and it has no grammar role.

So, I still struggle with this.  
I think I somehow understand when to use が or when to use は, but I have
trouble when I find it in long sentences or in the middle of them.

I understand why it cannot be used in a relative clause, and it makes sense to
me, but when I find it in the middle or at the beginning of a long sentence I
am not sure how to look at it and I get confused.

In some books I found sentences like:

私は[町に行く]人々に会った。 So it is part of the sentence.  
[私は][町に行く人々に会った] So は is not part of the sentence.

Somewhere I read that I should simply look at what is marked by は as if it
were connected to the main verb (so the last verb in the sentence).  
Is this correct?

Which is the correct explanation?
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grammar particle-は
は topic particle. What does it mean?
This article might help you:
<http://www.nkc.u-tokyo.ac.jp/study_info/study_info01_04_j.html>

I believe は has a grammatical role, as it can make a subject or an object, you
know.

> Somewhere I read that I should simply look at what is marked by は as if it
> were connected to the main verb (so the last verb in the sentence).

I think this is almost correct, because a topic is what you really want to
focus in a sentence and it is often said by the main verb of a whole sentence.

So when you find は at the beginning of a long sentence, mark it as the topic.
Its grammatical role (subject/object) is determined by checking the rest of
the sentence. Examples:

> 日本は去年結婚した私の息子が住んでいる国だ。
>
> Japan is the country where my son who married last year is living.

In the above sentence, the topic is Japan. So the speaker's intent is to tell
listeners additional information about Japan rather than information about his
son.

Grammatically, the rest of the sentence [去年結婚した私の息子が住んでいる国だ] is a verb phrase
([去年結婚した私の息子が住んでいる] modifies [国だ]) and it does not require an object, so Japan
becomes a subject.

> 数学は山田先生が教えている。
>
> Mr. Yamada teaches mathematics.

In this example, the topic is mathematics. So the interest of listeners is who
teaches mathematics, not what Mr. Yamada does.

Grammatically this は makes an object since the verb phrase [山田先生が教えている] needs
an object.

Note that the case exists that は is neither a subject nor an object:

> 象は鼻が長い。
>
> Elephants have a long trunk.

In the above sentence, the verb is [長い]. It doesn't take an object, so [象は] is
not an object. [象は] is neither a subject because if [象は] was a subject, the
meaning would be like "Elephants are long."

There is a discussion among Japanese about this kind of sentences. 三上章 (Mikami
Akira) persists that it has no subject.

Next, if you find find は at the middle of sentence, it can usually be moved to
the top of sentence without changing its meaning unless it is contained in a
relative clause. So you shouldn't be so scared; it is just indicating a topic
too.

> 目まぐるしい人工知能の進歩により将棋は人間の手が届かないものになってしまった。
>
> The rapid advancement of artificial intelligence has made Shogi what the
> human cannot reach.

The topic of this sentence is Shogi, which is a subject here. Its meaning the
same as [将棋は目まぐるしい人工知能の進歩により人間の手が届かないものになってしまった。], tough the former is a bit
more natural to me.

Hope my answer is your help.

# Edit (in response to the first comment)

> what if I find first a には、では、とは or even にとっては、に対しては、によっては、 and after that a
> normal は?

You know, は has major two meanings: one is a topic maker we are discussing
here, and the other is a contrastive particle.

If you find two は-s in one sentence, it is unlikely that more than one は-s are
topic makers. This is because it is quite unnatural that a sentence has more
than one topics.

> 実際には質問の文はなくてもかまいません
>
> Actually, the sentence in question can be omitted.

In this sentence, the first は in には is for contrast; the writer says that
"Actually the sentence can be omitted _though it isn't here_." the second は is
a topic maker.

I think particle+は is very much likely to be contrast rather than a topic. I
couldn't find any example in which particle+は is a topic maker.

So it is usually OK to take such words as contrastive ones.

The same applies to words like に対しては, によっては and others:

> やり方によっては彼は勝てるかもしれない。
>
> He might have a chance to win in some way. ( _by some particular method, not
> by the others_ )

Contrastive は is sometimes close to emphasis. In such cases は restricts the
meaning of sentence into a particular region as in the sentence below:

> その子は勉強が得意だが、数学者としては未熟だ。
>
> The boy is good at studying, but he is still inexperienced as a
> mathematician.

In this sentence, は emphasizes the phrase [数学者として] by separating
mathematicians and all the others. So the sentence implies that the speaker
does not do any evaluation on the boy out of the domain of mathematicians.
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は usually indicates the topic of the sentence.  
The topic of the sentence is what the sentence is talking about, which is not
necessarily the subject.  
Let's see an example:

コーヒーは毎日飲んでいます。kōhī wa mainichi nonde imasu.  
Speaking of coffee, I drink it every day.

When I say "コーヒーは", it means that I am going to talk about coffee.  
As long as I don't change the topic, all subsequent sentences will all be
about coffee. But coffee may not necessarily be the subject of the sentence.

Speaking of coffee,  
I drink it every day  
I love it,  
I think it is delicious.

> コーヒーは (kōhī wa)  
>  毎日飲んでいますし、 (mainichi nonde imasu shi)  
>  大好きだし、 (dai suki da shi)  
>  とても美味しいと思います。 (totemo oshii to omoimasu)

Coffee is not the subject of the sentence. But all sentences after コーヒーは talk
about coffee. Coffee is the topic of the sentences.  
The subject is the pronoun "I" which can be omitted because it is obvious who
the subject is.

Another example:

Speaking of Mr. Tanaka,  
his house is tiny  
but his dog is enormous.  

> 田中さんは (Tanaka san wa)  
>  家がとても小さいけど、 (uchi ga totemo chiisai kedo)  
>  犬が凄く大きい。 (inu ga sugoku ookii)

Mr. Tanaka is neither the subject nor object of the sentences. But we are
talking about his house and his dog, so he is the topic of the sentences.  
The particle は has nothing to do with the subject or object of the sentence.

Both sentences below are acceptable:  

> 田中さんの犬が大きい。(Tanaka san no inu ga ookii) Mr. Tanaka's dog is big.  
>  田中さんは犬が大きい。 (Tanaka san wa inu ga ookii) Speaking of Mr. Tanaka, his dog is
> big.

<http://hcm9999.blogspot.jp/2015/09/what-is-difference-between-and.html>
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