There are four bracket types that ought to be mentioned in the context of academic writing. To begin with, we have the parentheses, i.e. ( ). Secondly, we have square brackets, [ ]. Thirdly, we have curly brackets, { }. Finally, something must be said about angle brackets, < >.

Parentheses: ( )

Generally speaking, parentheses (singular: parenthesis), which can also be called 'round brackets', 'open brackets', or just 'brackets' in British English, are used to separate parenthetical (i.e. non-essential) information from the main text.

(1) This example (which was written in January) is intended to illustrate the use of parentheses.

Such parenthetical information, which can be of different types, is not necessary in order to understand the text, and a sentence that contains parenthetical information must be complete and understandable even if the parenthetical information is removed (this does not mean, by the way, that the sentence must make sense if only the actual parentheses are removed).  If we remove the entire parenthesis from (1) above, we thus get the acceptable and understandable (2):

(2) This example is intended to illustrate the use of parentheses.

In this particular case,  (2), we get an acceptable result if we just remove the parentheses (i.e. the symbols) too. However, in other cases, for instance in (3), the result of just removing the symbols would be strange, ungrammatical, or even incomprehensible.

(3) When you use parentheses (parenthes should not be used too often, by the way), do not forget to close them.

Parenthetical information is typically extra information that makes it easier for the reader to understand the text, offers a clarification, or gives the interested reader a fuller picture or some food for thought.

(4) Many people who use parentheses (or brackets, as they are called in less formal British English) should consider using footnotes instead.

In this respect, using parentheses is fairly similar to using footnotes (disregarding, of course, footnotes or endnotes that contain nothing but a reference to a source).   Parentheses are sometimes used to enclose examples of what the main text is discussing. Here, too, we can remove the whole parenthesis without causing ungrammaticality or serious comprehension difficulties, but the examples within parentheses are often very helpful to the reader. (5) is an example of this type of parenthetical information:

(5) The Baltic states (Estonia, Latvia and Lithuania) used to be part of the Soviet Union.

Sometimes the writer wants to add a comment of her own. Such a comment can be parenthetical in relation to the main message of the text, that is, it does not add any extra information about the subject matter as such, only the writer's personal opinion or reflections. 

(6) People who use too many parentheses and footnotes in their texts (I have to admit that I am one of them) are usually perfectionists.

In written documents of various types, parentheses are also used around numbers and letters in connection with various lists, numbered examples in a text such as this one, etc. Here is an example of what this may look like:

(7) Before we reach a conclusion, we have to take into consideration (i) his age, (ii) his CV, and (iii) his reputation.

In academic writing, parentheses are also used (in some reference systems) both when we refer to our sources in the text and in our reference lists or bibliographies. Parentheses are also used to enclose cross-references of the type found in (8):

(8) Parentheses also enclose cross-references (see chapter 5).

For more detailed information on how parentheses should be used in a particular reference system, please see the AWELU section on references and/or the reference guide or style manual that you have received from your teacher, supervisor, department, editor, or publisher.

Finally, please note that it is common practice not to capitalise the first letter of a sentence within parenthesis, or use a full stop at the end, when this parenthesis is inserted inside (or at the very end) of another orthographic sentence (see definition of 'orthographic sentence' below), as in (9).

(9) The use of brackets (this is the topic of this AWELU section) may be a problematic area for the inexperienced writer.

Definition of 'orthographic sentence'
"A sentence according to the normal conventions of written language: i.e. a minimal segment of text which begins and ends with sentence-boundary markers (normally this means beginning with a capital letter and ending with a full-stop" (Lancaster University, glossary, n.d.)
If, however, the parenthesis is not part of another orthographic sentence, normal punctuation rules apply, as in (10).
(10) The use of brackets may be a problematic area for the inexperienced writer. (This is the topic of this AWELU section.)
In line with this, if there is more than one sentence within the same parentheses, a full stop is used to separate them.

Square brackets: [ ]

An important use of square brackets in academic writing is when we want to add something to, or somehow change, emphasise, or comment on a direct quotation from some other source. There are a number of different cases of that sort that will now be mentioned and exemplified. Sometimes we want to add a piece of information in order to help the reader better understand the quotation.
(11) According to Harris (2005:63), "the Scandinavian countries [Denmark, Norway and Sweden] have sound economies."
Sometimes we have to change the capitalisation or add a pronoun in order for the quoted material to be grammatically appropriate or understandable in its new context. Then we use square brackets to indicate where such changes or additions have been made, as in (12) and (13):
(12) According to Harris (2005:63), "[t]he Scandinavian countries have sound economies." (13) According to Harris (2005:63), this is partly due to "[their] being rich in natural resources."
According to some reference systems (e.g. MLA), square brackets should be used to enclose the three dots that symbolise ellipted material in quotations, to indicate clearly that these ellipsis dots were not present in the sentence originally. It looks as in (14).
(14) According to Harris (2005:63), "the Scandinavian countries [...] have sound economies."
If you should find it necessary to point out that what you have written in a quotation was actually there in the original text, you can write [sic]. Please note that the word sic is not an abbreviation (so no full stop after it) and that sic (but not the square brackets) should be italicized, but not underlined, as in (15).
(15) According to Harris (2005:63), "the Scandinawian [sic] countries have sound economies."
Square brackets can also be used when you want to point out to the reader that you have added italics, underlining, boldface, etc. in order to emphasize some part of the original text. It could look as in (16).
(16) According to Harris (2005:63), "the Scandinavian countries, including Iceland, have sound economies." [my emphasis]
We can also use square brackets if we want to insert parenthetical information inside another parenthesis, as in (17):
(17) In his recent work (which has [unfortunately] not been published yet), Professor Watson discusses the usefulness of square brackets.
For obvious reasons of readability, we should not overuse this possibility. You can read more about how to edit quotations if you follow this AWELU link:Follow the link below to see more about how brackets are used in reference lists and in academic texts when we refer to our sources.Finally, note that there are of course many other technical uses of square brackets, for instance to denote an interval in mathematics, or to enclose phonetic script or syntactic consituents in linguistics.

Curly brackets: { }

Most academic writers can go through their entire careers without ever needing to use curly brackets (also known as [curly] braces). However, within certain fields of research, curly brackets fulfil specific important and conventionalised functions. For instance, curly brackets are used in mathematics, logic, and linguistics to enclose members of an unordered set, as in (18).
(18) E = {2, 4, 6, 8, ...}
Curly brackets are also frequently used in various ways in many computer programming languages. AWELU is not the right place (if there is one) for a complete list of all different uses of curly brackets (or any other type of bracket) in academic texts.

Angle brackets: < >

Just like curly brackets, angle brackets are frequently used in many technical contexts, for instance in computer programming. Each writer and programmer had better learn the conventions within his or her particular field. We should all be aware, however, of the use of angle brackets to enclose e-mail addresses, IRLs (Uniform Resource Identifiers) and URLs (Uniform Resource Locators, i.e. web pages) to separate them from surrounding text, especially when there is a risk for ambiguity. An example of the use of angle brackets in connection with web pages is found in (19), which is taken from the AWELU section on documentory note styles (see the link below the example):
(19) The first time a source is referred to, a full reference is provided in the note:
1. Maria Colenso, "Meerkat habitat and diet", para. 4, How Stuff Works (13 May 2008). <>, accessed 22 Jan. 2011.
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