Doc Fee Excerpted and adapted from a shiur by Dayan Yehoshua Grunwald September 21, 2023…
Excerpted and adapted from a shiur by HaRav Chaim Weg
February 16, 2023
When you will encounter an ox of your enemy or his donkey wandering, you shall surely return it to him.
If one finds a lost object with no simanim (characteristics by which the owner could identify it), we assume the owner gave up hope of getting it back (yeiush) and relinquished ownership, so the finder may keep it.
If it bears simanim and was found in a majority-Jewish location, the mitzvah to return it (hashavas aveidah) applies. One must take the object and publicize what he found (hachrazah) to locate the owner. He has the status of a shomer (custodian) and is liable if the item is damaged. The Shulchan Aruch (C.M. 267:16) cites a dispute whether he is a shomer chinam (unpaid custodian) or a shomer sachar (paid custodian, with greater liability).
If no one claims the item for a few weeks, the finder must hold it for the owner ad sheyavo Eliyahu (until Eliyahu comes). R’ Moshe Feinstein and other contemporary poskim say that if the item is easy to replace, the finder may record the simanim and sell or keep it, and when Eliyahu identifies the owner, the finder will pay him or replace the item. An item that would have sentimental value to the owner should be retained.
 If the owner has not yet discovered that the item is missing (yeiush shelo mida’as), Amora’im disagree whether it is already ownerless; the halacha is that it’s not. R’ Akiva Eiger holds there is no obligation to return such an item, but R’ Moshe Feinstein disagrees.