If you're a teacher in Moodle, then this page will tell you about the major changes affecting you in Moodle 3.7
Messaging: Delete conversations in groups
Teachers may be given a capability to delete messages from users in group conversations.
Forums: Reply privately
Teachers or others with permission can add a private reply in a forum, visible by only the recipient.
Badges: Competencies as a criterion
Badges may now be awarded based on competencies, (and competencies may be graded when grading activities)
Course custom fields
If enabled by admin, custom fields in the course settings may be added to provide extra information and improve searching.
Teachers can view hidden chapters in a book with the editing turned off .
Lesson: Include media when grading essays
The Lesson activity now allows for teachers and students to include media in Essay type questions.