Their vs. There

Their (determiner)

Belonging to, from, of, or relating to, them (plural).

“they will meet tomorrow at their convenience;”

“this is probably their cat”

Their (determiner)

Belonging to someone (one person, singular).

Their (adverb)

misspelling of there

There (adverb)

In a place or location (stated, implied or otherwise indicated) at some distance from the speaker (compare here).

There (adverb)

In that matter, relation, etc.; at that point, stage, etc., regarded as a distinct place.

“He did not stop there, but continued his speech.”

“They patched up their differences, but matters did not end there.”

There (adverb)

To or into that place; thither.

There (adverb)

Where, there where, in which place.

There (adverb)

In existence or in this world; see pronoun section below.

There (interjection)

Used to offer encouragement or sympathy.

“There, there. Everything is going to turn out all right.”

There (interjection)

Used to express victory or completion.

“There! That knot should hold.”

There (noun)

That place.

There (noun)

That status; that position.

“You get it ready; I’ll take it from there.”

There (pronoun)

Used as an expletive subject of be in its sense of “exist”, with the semantic, usually indefinite subject being postponed or (occasionally) implied.

“There are two apples on the table. [=Two apples are on the table.]”

“There is no way to do it. [=No way to do it exists.]”

“Is there an answer? [=Does an answer exist?]”

“No, there isn’t. [=No, one doesn’t exist.]”

There (pronoun)

Used with other intransitive verbs of existence, in the same sense, or with other intransitive verbs, adding a sense of existence.

“If x is a positive number, then there exists [=there is] a positive number y less than x.”

“There remain several problems with this approach. [=Several problems remain with this approach.]”

“Once upon a time, in a now-forgotten kingdom, there lived a woodsman with his wife. [=There was a woodsman, who lived with his wife.]”

“There arose a great wind out of the east. [=There was now a great wind, arising in the east.]”

There (pronoun)

Used with other verbs, when raised.

“There seems to be some difficulty with the papers. [=It seems that there is some difficulty with the papers.]”

“I expected there to be a simpler solution. [=I expected that there would be a simpler solution.]”

“There are beginning to be complications. [=It’s beginning to be the case that there are complications.]”

There (pronoun)

That.

“therefor, thereat, thereunder”

There (pronoun)

Used to replace an unknown name, principally in greetings and farewells

“Hi there, young fellow.””

There (determiner)

misspelling of their