Is it okay to record audio for a lecture? Sometimes the Professor will say things quickly, or say things before I can write them down. It would be of great benefit if I could record audio for the course, but I am not sure the legal guidelines around doing so.
The legal issues involved will, of course, depend on what legal jurisdiction you are asking about. However, there are some common factors that can be considered.

In the UK many institutions take the interpretation of the Disability Discrimination Act (DDA) that teaching staff cannot prevent students making private recordings (Audio or Video) of any teaching activity. The reasoning is that students are entitled to keep their condition confidential (as it may be a private medical matter) but also must be permitted to receive the appropriate adjustment for their condition. It would be expected, as a courtesy, that the student would inform the staff concerned that recording was taking place, but that this could not be enforced. (*)

Conversely, the Intellectual Property (IP) of the class and the material contained therein would be held by the University and the individual teacher to varying degrees. The recording could not be published or used for any means other than private study as this would then violate the rights of the owner of the IP.

Many aspects of disability discrimination, privacy and intellectual property legislation is common across many EU states, but there are variations in practice. I can imagine how these different statutes might interact in (say) the US, but I suspect it might vary from state to state and not always be a federal matter, and so I decline to speculate.

(*) I have responsibilities for assisting academic staff understand these issues in my department...