DB structure: users, groups & permissions

[Rom Grk]

Hello,

I would like some advise on the following subject.

I have 2 data models:
- Users:
ID, name, permissions & group
- Groups:
ID, name, location, etc.

Currently, 1 user can only belong to 1 group.
Now, I would like to update this so that one user can belong to multiple groups. So my initial idea was to change the User.group field to accept multiple groups instead of just one. However, I do not feel that this is a good idea.
Should I create a table for each group, and in that table have a list of users, or should I go with the list-field option?

Also, I have a similar questioning in a forum-like section that I have to build. Any advice on how I should structure this DB?

Thank you for your time.