If your site used to send its emails through the Gmail SMTP server and stopped after the latest Joomla update, you can restore the sending by modifying one of Joomla files.  Modifying core files is not recommended but sometimes it has to be done as an emergency measure, until a permanent remedy has been implemented in the applications involved.

Versions of Joomla before 3.6.0 used to work with Gmail by using an app password, created from the 2-Step Verification page of Google Accounts at https://security.google.com/settings/security/apppasswords

After the update to version 3.6.0, the connection to smtp.gmail.com reports a failure in the verification of the server certificate:

[28-Jul-2016 09:54:34 UTC] PHP Warning: stream_socket_enable_crypto(): SSL operation failed with code 1. OpenSSL Error messages:

error:14090086:SSL routines:ssl3_get_server_certificate:certificate verify failed in C:\www\joomla360\libraries\vendor\phpmailer\phpmailer\class.smtp.php on line 343

As instructed under "PHP 5.6 certificate verification failure" at https://github.com/PHPMailer/PHPMailer/wiki/Troubleshooting, you can set the Options parameter to bypass the verification of the server certificate.

The following code can be added as a workaround to the beginning of the function useSmtp() in the file libraries/joomla/mail/mail.php:

   // 20160729 workaround for certificate verification failure - ref. https://github.com/PHPMailer/PHPMailer/wiki/Troubleshooting
    $this->SMTPOptions = array(
    'ssl' => array(
      'verify_peer' => false,
      'verify_peer_name' => false,
      'allow_self_signed' => true
    // 20160729 end