What do all languages have in common ? I'm looking for a list of features (such as grammatical, semantic or phonetic elements) that are present in all natural languages.
Any linguistic answer to this question has to be at least partly theory laden. There are many approaches to linguistic universals.
The most general points would be: 1. All natural languages can be acquired by people born into a community of speakers, or learned by people as a second language (with well-known limitations). 2. The propositional content of all natural languages can be translated from one another given limitations of vocabulary and contextual understanding. 3. The non-propositional content of all languages can be conveyed between one another through some means of human communication.
From the above it follows that all languages will have the same physiological, cognitive, and neural underpinnings (whatever those may be).
However, that seems quite trivial. So when people ask those questions, they are generally looking for some feature like nouns, vocabulary, syntactic constraints. Here the people making the strongest claims are:
- Universal Grammar which claims that all languages share certain constraints on their syntax which are the only way to explain their learnability. However, there are only a handful of these and those are virtually incomprehensible. This is a position most often associated with Chomsky and most popularly explained by Steven Pinker.
- Semantic Primes claims that there are about 60 words that are common to all languages and through which all meaning can be described (they call this Natural Semantic Metalanguage). It is based around the work of Anna Wierzbicka and the group does some really interesting research even if their broader claim is a bit less palatable.
- Modern linguistic typology (whose perspective was taken on in @Darkgamma's answer) makes much weaker claims few of which stand and fall with a single counter example. The most accessible explication of this position is RMW Dixon's 'Basic Linguistic Theory' which, while not always most uptodate on all issues, gives an accessible overview (over three volumes). Historically, this work on universals was associated with Joseph Greenberg but now the project is much less interested in universals than common patterns and tendencies.
- Historical linguistics also makes some claims about universals mostly to do with common origins. The strongest claims are made by some branches which try to find common origins of all languages, e.g. Nostratics, who are not necessarily taken seriously by the mainstream. But even more traditional historical linguistics makes some claims about fairly universal principles of language change.
By the way, both 3 and 4 sometimes call themselves comparative linguistics, so it's sometimes hard to know what is meant by the term without some context.
Personally, I recommend that every linguist or even anybody interested in language spends some time with modern linguistic typology (which has moved far beyond the inflectional/agglutinting business). While Dixon's 'Basic Linguistic Theory' may not be the first thing on the list, it should be required reading for any linguistics graduates long before they read anything by Chomsky.
Finally, The World Atlas of Linguistic Structures (WALS) http://wals.info is a great place to check any claims as to universality, even if it (unavoidably) relies on data of uneven quality. But it also nicely illustrates the extent and the limitations of the current evidentiary base for claims about universals.
Well, the basics are the same: all languages have consonants and vowels, and always more consonants than vowel qualities. All of them have verbs and, slightly controversially, all of them have nouns. The reason for the controversy is that some languages have nouns that look and behave a bit like verbs.
All languages have syntax, the core words for body parts, the basic colour terms for "black" and "white", at least for "parents" and "children". Words for "night", "day", "moon", "rain" etc.are also universal. All languages have pronouns of some kind, at the very least first person and non-first person. All languages have a way of expressing possession.
While I can't cite sources, it is often believed that all languages have at least some rudimentary morphology: even Mandarin Chinese has a derivational suffix. Furthermore, all languages have a strategy they use to distinguish agent from patient, as well as some other thematic relations.
There is probably more, but these are the basics that all languages (I think) share.