I would like to export email from the Mac OS X Mail app to Outlook and the other way round. How do I do this?

1. You can try OLM to PST Convert a third party tool.
   This software Features
   Provides user with the safe and reliable platform to convert all user mailbox data stored in OLM file
   Restores damaged or corrupted OLM file instantly and extracts data from it
   Make use of advanced functionality to display the preview of user data after extracting them from OLM file
   Facilitates user with exceptional option to stop the running process of generating items from recovered folder if required
   Provides user with an option to save the recovered folder items as HTML file
   Competent enough to perform the successful conversion of Outlook for Mac emails to MS Outlook
   For more information and free download here : http://www.olmtopst.n.nu

2. Outlook for MAC stores data in .olm format wher as Outlook for Windows store data in pst format. And there is no way to import olm file to Outlook for Windows. For this you have to convert your olm files into pst format with some tool and then import pst file into outlook. Searching google for olm to pst converter tool i got link to this tool Kernel for OLM to PST converter which says that using this tool you can convert olm file into pst file.

3. i use outlook at work, and frequently need to move my mail to my mac, which is running mountain lion os x. i basically export any new emails from my PC using 8Convert (www.eighthoof.com), then import it directly into Apple Mail.

   i haven't had to move mail back the other way (from my mac back to outlook). let me know if you find a good solution for that..

4. To import emails from Mac Mail to Outlook:
   You may extract emails from Mac Mail app using this free tool (for Windows):
   http://www.outlookconverter.com/downloads/macmailemailexport.exe
   Or the free Email Extractor tool for Mac OS:
   http://www.outlookimport.com/free-email-extractor/
   After that you may use any other tool to import extracted emails to Outlook. Or just open them if your Outlook version is 2007 or 2010.

   To export emails from Outlook to Mac Mail you may do the following:
   1. Use the Outlook Export Wizard from http://www.outlookexportwizard.com to save emails contacts and other Outlook items to files.
   2. Transfer extracted emails to Mac OS computer or connect to the folder containing EML files via network
   3. Run Mac Mail
   4. Select portion of EML files
   5. Drag them into desired Mac Mail folder.
   Your emails will be imported automatically.

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6. The easiest option probably is to use an IMAP or Exchange service. If that's not possible Thunderbird is the best option as @ha14:disqus already suggested.

   Some detailed guides:

   Outlook to Apple Mail (using Thunderbird)
   http://www.howtogeek.com/howto/22285/import-email-from-outlook-to-apple-mail.app-using-thunderbird/

   Apple Mail to Outlook (via emlx or Outlook Import Wizard)
   http://www.outlookimport.com/export-e-mails-from-apple-mail-and-import-to-ms-outlook/

   Apple Mail to Outlook (using EML to PST Converter)
   http://www.emltopst.com/apple-mail-outlook-import-eml/

7. Steam vr on mac. How to Transfer Emails From Outlook to Apple Mail
   http://www.trickyways.com/2010/08/how-to-transfer-emails-from-outlook-to-apple-mail/

   Export From Outlook 2011 Mac & Import .eml Files Into Apple Mail
   http://marcushesse.com/2010/exporting-mail-from-outlook-2010-mac-importing-eml-files-into-apple-mail/

   O2M shareware
   http://www.littlemachines.com/

Launch Outlook for Mac 2011, click File menu and choose Export in the context menu. On the Export wizard, select Outlook for Mac Data File (.olm) option and Items of the following types: Select the items to export, i.e. Mail, Contacts, Calendar, Tasks, and Notes, and then click Next button. If you already have a copy of Outlook for Mac, simply launch it, choose File Import, select Outlook Data File (.pst or.olm) in the first Import window, click the Next button, select Outlook for Windows Data File (.pst), click Next again, navigate to the.pst file you’ve copied to your Mac.