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Is it posible that you could give us a little bit more information;
What Plan are you on?
Is it the same when using Outlook Web Access?
What version of Outlook are you using?
How is your Outlook setup (Exchange, POP or IMAP)
Does the email leave the Outboox and appear in Sent Items?
If on an E plan (or a P1 plan congiured without NS record change), have you added the SPF TXT record at your host registrar?
What is your custom Domain name (so that the DNS records can be checked)?
Do you recieve any Non Delivery Reports?
Is this the same for sending emails internally (to other users on 365)?
Thanks for posting in the forum.
If you could fill out the answers to the questions so we have moe information to determine what causes your issue. The additional information will help us to identify the root cause of your issue and lead to a timely solution.
We still have not received the answers to the questions that were asked to further assist you. With the information returned we will be able to assist you to resolve this issue.
If you have any additional questions or concerns, please let me know and I will continue to assist as needed. I look forward to hear from you soon.
Sherika MSFT Support