CiviCRM is a great open source AMS
CiviCRM has a strong track record of serving associations since 2005, with over 10,000 installations today around the world in a variety of languages.
It has a flexible system for memberships and event registration, and integrates with numerous payment processors. It integrates with popular accounting packages like QuickBooks, Sage and Great Plains. For associations lthat need them, there is support for managing grants, volunteers, cases and campaigns. Custom fields are intuitive and powerful. Bulk email and bulk texting is supported, as well as social media integration. There are hundreds of extensions providing enhanced functionality and integration with external services. For charities, it has extensive fundraising options.
For many associations, the excellent integration with their website will be crucial. WordPress, Drupal, and Joomla, are amongst the CMSes supported, together accounting for over 2/3 of websites using CMSes.
JMA Consulting provides CiviCRM Consulting services to many kinds of associations, including regulated professionals, business associations, disease support organizations, and other associations with advocacy and fundraising needs.