Is lastpass.com Safe?

I am concerned about the needless requirement that all that info must be placed in the vendor's site. I would prefer that the encrypted data be kept where I need or want it, and decrypted only locally. This would also allow me to use it off-line to access my local encrypted files. Allow me to back it up as any other encrypted file under my control, without dependence on their corporate integrity.

There are some questions about Lastpass since the company was purchased by a company with a questionable record.  In my experience, it remains one of the most-integrated, and securely structured, password managers.  The ability to use second-factor authentication, especially the secure and easy to use Yubikey, is a great addition.