Turkish/Converbs

Converbs are a feature that is quite unique to Turkish grammar. If your native language isn't a Mongolic or Tungusic language, relating ideas through converbs should be pretty new to you.
Converbs allow us to use subordinate clauses adverbially. While English uses conjunctions for this purpose, in Turkish, the verb receives a special ending instead. Compare the following examples:
While Turkish does have conjunctions, they are used way less than converbs, and sentences with converbs sound better in most cases. Some meanings on the other hand may only be achieved through converbs, and as such, learning converbs is fundamental to speaking Turkish fluently.
Simple converbs.
Simple converbs may only be attached directly to verb stems, and are not declined for person. The subject of the subordinate clause is thus implied to be the same as the main clause unless explicitly stated. The following suffixes are the most common ones (all suffixes below undergo vowel harmony):
"-(y)ıp".
"-(y)ıp" is a suffix that connects two equivalent sentences, and can be thus loosely translated as "and".
When a verb is suffixed with "-(y)ıp", it is assumed to have the same tense markers as the main verb, and the subject of the "-(y)ıp" clause is always the same as the main clause.
Thus, "-(y)ıp" can be replaced by the conjunction "ve" in most cases.
When the main clause is affirmative, the clause with "-(y)ıp" can take the negative suffix "-ma" if it is negative.
However, if the main clause is negative, the subordinate clause is also implied to be negative, and thus can't receive the negative suffix "-ma".
When the main clause is negative, but the subordinate clause is not, the word "da/de" is used to emphasise this.
When combined with the verb "durmak", it means "to keep doing something". 
The verb may be doubled to stress this.
"-(y)ınca".
The suffix "-(y)ınca" corresponds closely to the English conjunction "when". It represents a sequential relationship between two events.
The sub clause with "-(y)ınca" may also have a subject different than the main clause. When this happens, the subject is in nominative, and no personal suffixes are added to "-(y)ınca".
"-(y)arak".
The converb "-(y)arak" is used to describe the manner an action is performed in, or the means used to achieve the said action. It roughly corresponds to the English preposition "by".
The subclause with "-(y)arak" almost never receives a subject other than that of the main clause. Very rarely, an item possessed by the subject of the main clause may act as the subject.
The verb "olmak", when declined as "olarak", has a special meaning, roughly corresponding to that of the English preposition "as". It may be used to produce adverbial clauses from adjectives...
...to show status or classification...
...or to show that a group is being involved in an action.
Both the suffix "-(y)ınca", and the suffix "-(y)arak" are stressed on the first syllable.
"-madan".
The suffix -madan means "without doing ..." and as such, it is always negative. The last syllable of the stem is always stressed in pronunciation, since it contains the negative suffix.
"iken".
"iken" is an exceptional converb, as it is appended to a declined verb, or a noun, similar to other copular suffixes. When it is appended to a word, it becomes "-(y)ken". It is important to note that the "e" in "-(y)ken" is invariable. It can be roughly translated as "while", as it is used to state an ongoing event that happens while the main clause is occuring.
When the suffix "-(y)ken" is attached to a verb, the verb references a time relative to the main clause. Moreover, "-(y)ken" is never declined for person, and works similar to "-(y)ınca" in that sense. Let's examine the different meanings produced by appending "-(y)ken" to different tenses.
Aorist or Present.
The most common way of using "-(y)ken" is in conjunction with the aorist tense. In this form, it closely correpsonds to the English "while". The present tense may also be used, albeit less commonly, with the same meaning.
Past.
The suffix "-(y)ken" may only combine with the "-mış" past tense, and never the "-dı" past tense. It is used to talk about events that were already complete while the main clause was occurring. Thus, it produces a meaning that can be roughly translated as "having done ...".
This meaning can be further stressed with the adverb "hazır".
Future.
When "-(y)ken" is used with the Future Tense, it specifies that the action had not yet occured when the event at the main clause occurred. Often, this also implies that the event did not get realized as it was supposed to.
Optative.
On less formal registers, "-(y)ken" may combine with the optative tense "-malı" to talk about obligations that were present when the event occurred. Similair to the future tense, this often implies that the obligation was not realized.
Nouns.
"-(y)ken" may also combine with nouns, "var/yok" and "değil" freely. 
Reduplicated Converbs.
Reduplicated converbs are special, since two verbs are used instead of one. There are two converbs with this form.
"-(y)a -(y)a".
There are two ways this converb may be utilized.
By using the same verb twice.
The most common form of this converb is used by repeating the same verb twice. This converb is a more stressed version of "-(y)arak", stressing the continuity or the repeatedness of an action.
By using two different verbs.
Sometimes, two different verbs may be used for this converb. The two verbs are usually either similar, or opposite in meaning. Forms with these construction are mostly idiomatic.
"-(a)r -maz".
This converb is formed by repeating a verb twice, once with the affirmative and once with the negative forms of the Aorist. This converb is translated as "as soon as" in English. 
Compound Converbs.
Compound converbs are formed by combining a declined verb with a postposition. 
"-mış" "gibi".
"-mış" here can either be the past tense suffix "-mış", or the copular suffix "-(y)mış". This produces a counterfactual meaning, similar to English "as if..." . It may be stressed with the adverb "sanki".
When combined with the verb "durmak", it means "to look as if ..." .
"-dıktan" "sonra".
This converb means "after ..." .
It may combine with pronouns in a similar way to "-(y)ınca".
"-madan" "önce".
This converb means "before ..." and functions very similar to "-dıktan sonra". However, since it contains the negative suffix "-ma", the last syllable of the stem is stressed in pronunciation.
"-ana kadar".
This converb means "until ..." and functions very similar to "-dıktan sonra" and "-madan önce".
"-dığından beri".
This converb means "since ..." and functions very similar to the converbs described above. However, as we will see in Turkish/Relative Clauses, "-dığından" is declined for person.
"-dığı için".
Similar to the converb above, this converb is declined for person, and will be covered in greater detail in Turkish/Relative Clauses. It is used to express cause, and is a better alternative to the conjunction "çünkü".
"-mak için".
This converb was already covered in detail in Turkish/Infinitives. It is used to express aim.