Ahead of the October release of macOS 10.15 Catalina and the first shipments of the new iPhone 11 line loaded with iOS 13 and Safari 13 for iOS, Apple has released Safari 13 for 10.14.6 Mojave and 10.13.6 High Sierra. The new version includes an updated start page with Favorites, frequently visited, and recently visited Web sites; lets you enable Picture in Picture from the audio button in a tab; and enables jumping directly to an already-open tab from the search completion list when you type the Web site address into the Smart Search field. Safari also now warns if you use a weak password when signing in to a Web site and offers to help upgrade to a stronger password, and it adds support for authentication using USB security keys on supported Web sites.
There have been a few early reports of problems with Safari 13 (see the comments) so you may wish to hold off for a bit on installing the update. Safari 13 is available only via Software Update. (Free, 10.13.6 and 10.12.6)