Which remote desktop cross-platform apps would you suggest?
TeamViewer, without a doubt. It's not that great for controlling iOS and Android, but other than that, it's as good as it gets. Free for personal use, and reasonable price for commercial use.
Thanks. Please tell , what are limitations of free version over commercial version of teamviewer?
The remote access functionality is identical (apart from the free version not being available for commercial use). However, the commercial version has additional features such as building up a cloud-based list of clients, the option to create branded clients, the ability to download and run the latest version automatically and more support options.
Those extra options are nice, but largely irrelevant to private users anyway.
Teamviewer supports Windows, OS X, Linux, Android, and iOS
Splashtop supports Windows, OS X, Linux, Android, and iOS