Microsoft are updating the requirements to access Office 365 services from late 2020.
In summary, from October 2020 you will need to be using a version of Office in mainstream support to access Office 365 services, this means Office 2019 or newer.
If you are using an older version of Office (2010, 2013 or 2016) you will only be access Office 365 through the website (e.g. you won't be able to use Outlook to access your email on your PC).
If you already get Office on a subscription basis as part of your Office 365 licence (Business Premium / Enterprise E3) you don't need to worry, you'll be automatically upgraded to the latest version of Office.
UPDATE Sept 2018: Microsoft have changed their mind, extending support of Office 2016 with 365 until 2023.
Office 2016 connectivity support for Office 365 services
In addition, we are modifying the Office 365 services system requirements related to service connectivity. In February, we announced that starting October 13, 2020, customers will need Office 365 ProPlus or Office 2019 clients in mainstream support to connect to Office 365 services. To give you more time to transition fully to the cloud, we are now modifying that policy and will continue to support Office 2016 connections with the Office 365 services through October 2023.