Perhaps Dorothy Turner at the University of Ottawa puts it best when she notes that:

A topic sentence (also known as a focus sentence) encapsulates or organizes an entire paragraph, and you should be careful to include one in most of your major paragraphs. Although topic sentences may appear anywhere in a paragraph, in academic essays they often appear at the beginning.

It might be helpful to think of a topic sentence as working in two directions simultaneously. It relates the paragraph to the essay's thesis, and thereby acts as a signpost for the argument of the paper as a whole, but it also defines the scope of the paragraph itself.

Topic sentences often act like tiny thesis statements. Like a thesis statement, a topic sentence makes a claim of some sort. A claim is an assertion. It is understood that, by its very nature, a claim is something that is arguable. In other words, people should be able to disagree with it. As the thesis statement is the unifying force in the essay, so the topic sentence must be the unifying force in the paragraph. Further, as is the case with the thesis statement, when the topic sentence makes a claim, the paragraph which follows must expand, describe, or prove it in some way. Topic sentences make a point and give reasons or examples to support it. Finally, topic sentences should "stand on their own." They should pass the "blank paper test." In other words, you should be able to print off just your topic sentence on a blank piece of paper and show it to people. Based solely on that once sentence, everyone should know precisely what the paragraph's perspective of the thesis statement will be about.

To be clear, the argument that is advanced by the topic sentence must be connected to or related to the thesis statement that was given in the introductory paragraph. In programming terms, it could be said that this is a “parent-child” relationship. The topic sentence is a “child” of its “parent,” the thesis statement. In hierarchical terms, it could be said that the topic sentence is subordinate to the thesis statement.

One tip that students can use to test this connection is to invisibly place a question response word or words such as “because,” “by,” or “that is to say” between the essay’s thesis statement and each of its topic sentences. Take the following examples:

In addition to connecting with the thesis statement, topic sentences must also respect the same rules a thesis statement must follow. In other words, they must be arguable, worded strongly, and stand on their own.

The 2nd, 3rd (etc.) topic sentences should start with either a transition word, a transition phrase, a connector, or a bridge. These words inform readers that the author is leaving behind one topic and moving on to the next one.