This is just a set of notes about the subject, not yet well organized and far from exhaustive.
Existence is the ontological topic par excellence. In Anglo-American philosophy (this article will have to be augmented with summaries of work in other traditions), probably the most widely-asked question about it is what sort of concept it is, or what function it serves in languages, both natural and formal. Another significant topic, related to the first, but somewhat less discussed, is whether 'existence' or 'exists' can be analyzed or defined or otherwise explicated, and if so, what the explication might be. (For further discussion, see the existence of physical objects.)
Hume and Kant, among others, are each well-known for their shared view that the claim that a thing exists, when added to our notion of a thing, does not add anything to the concept. For example, if we form a complete notion of Moses, and superadd to that notion the claim that Moses existed, we are not adding anything to the notion of Moses. [This needs to be greatly expanded.]
Frege and Russell, among many others, for similar reasons are well-known for their view that 'exists' is not a (logical) predicate, or more precisely, not a first-order predicate--or, perhaps equivalently (and perhaps not), that existence is not a property. This has become the dominant but not the universal view in twentieth-century and contemporary Anglo-American philosophy. But, as G. E. Moore pointed out in an early essay, it is a matter of some difficulty to say what exactly what this view amounts to. [This too needs to be greatly expanded, and probably made into its own article.]
The words (and concepts) 'existence' and 'being' are treated in slightly different ways in Western philosophy. Aristotle pointed out that there are various ways in which a thing can "be" and inaugurated ontology as a field with his notion that there are categories of being, such as substance, attribute, and acting-upon. Similar claims, however, are not as often made on behalf of existence. That is, contemporary philosophers at least are wont to treat existence as a univocal, unambiguous concept, as if the only sense of 'existence', or the only sort of existence worth talking about, were the existence of physical objects. Consequently, some discussions of existence have an unclear bearing on, for example, the sense in which numbers, possibilities, and properties exist (or might be thought to exist).
Even if the ambiguity of 'exists' is sometimes overlooked, oddly enough, the ambiguity of 'does not exist' is not. That is, ontologists are fond of pointing out that there are various ways in which things can fail to exist: they can be fictional (Sherlock Holmes), imaginary (the golden mountain), legendary (Loch Ness Monster), mistakenly inferred (the ether), etc. The multiplicity of ways in which a thing can fail to exist has be so striking to some that it has been suggested that existence is, in fact, merely an "excluder" concept--used to classify items by what they are not (not fictional, not imaginary, not mistakenly inferred, etc.)--as 'real' is sometimes thought to be.
Though often not discussed under the heading of existence, disputes among realism, phenomenalism, physicalism, and various other metaphysical views concern what might be called the criteria for existence. For example, phenomenalism, generally speaking, is the view that everything that exists is mental. Most phenomenalists would want to deny that this claim is a definition of 'exists'; if phenomenalism were treated as a definition of 'exists', then others might accuse the view of trying to be "true by definition." Accordingly, it might be dismissed as a trivial exercise in redefining the ordinary concept of existence, which is, perhaps, of little interest to anyone. Exactly what relation, however, definitions (or analyses, or explications, etc.) and criteria have is an interesting and vexed question. See definitions vs. criteria.