When writing papers, I meet typos/grammatical errors in quotations now and then. Should I correct them, or leave them as they are?
When quoting, the convention is to quote the text exactly as it appears in the source. It is common to mark a misspelling (or other problematical word) that might be otherwise thought to be an error by you (the quoter) by following it with "[sic]". The word sic (which is from Latin) means "as such", that is, the apparent problem occurred in the original text. The square brackets indicate that the text was added by the quoter. For instance,
Our massage treatments help relive[sic] your pain.
Adding corrective braces can be useful for this.
Instead of "He go into the sunset" you could gently correct with "He [goes] into the sunset" .
Instead of "I'm going to do it careful" you can use "I'm going to do it [carefully]."
You can also use "[sic]" if you want to leave the error present, and highlight its presence.
It's also common for brackets to be used to modify the tenses and pronouns of the quoted material to match the new text into which it is nested.
Here's example involving verb tense, from the 'net:
The judge's order "restrain[ed] and enjoin[ed] the further implementation" of the law, including the prevention of Secretary of State Doug LaFollette (D) from publishing the act in the Wisconsin State Journal. TPM
The original had present tense, which was altered by the brackets in the second version:
"I do, therefore, restrain and enjoin the further implementation of 2011 Wisconsin Act 10," Sumi said, according to a transcript. Wisconsin State Journal
The adverb sic—meaning "intentionally so written"—when added just after a quote or reprinted text indicates that the passage is just as it appears from its original source. The usual purpose is to inform readers that any errors or apparent errors in the copied material are not from transcription—i.e. that they are reproduced exactly from the original writer or printer. Sic is generally used inside square brackets, like [sic], and occasionally parentheses/brackets: (sic).