If you're building the software on a non-Microsoft platform, you may be able to use the dll for the Client Object Model via Mono (or an equivalent?), but I wouldn't necessarily trust it to be stable. I know Mono works best with dlls compiled through Mono.
SharePoint offers a variety of web services to interface with, where you can perform a number of common tasks. Just be aware of the restrictions of Office 365, as you have less privilege on the cloud platform, and a significant amount of functionality is restricted. Office 365 is fairly locked down.
To get started with using the SharePoint web services, check out MSDN's page "SharePoint 2010 Web Services"
Additionally, if you'd like to go down the Client Object Model path, here is the MSDN page: SharePoint 2010 Client Object Model