However, accessing your computer from outside the home or office is more complicated. You'll need to configure your router so that the relevant ports are passed through to the computer you want to connect to. You'll find guides online. It's also wise to configure a dynamic DNS service so that you can connect via a hostname rather than an IP address, which is prone to changing. Examples of dynamic DNS services include NoIP and DuckDNS, which are free of charge, but how they're configured is again outside the scope of this article. There's many guides available online - just search using your router model number and "dynamic DNS".
Setting-up your Mac and iPad for screen sharing
Mac OS X uses the established Virtual Network Computing (VNC) protocol to share a desktop remotely, so any iOS app that supports VNC will work. There are quite a few of these, in fact, but one of the best that also happens to be free is VNC Viewer. It can be found in the App Store but before using it you'll need to configure the Mac you want to connect to: put a tick alongside Screen Sharing in the Sharing component of System Preferences, then click the Computer Settings button and ensure there's NOT a tick alongside either of the two headings you see.
Note the address listed beneath the heading that reads Screen Sharing: On. You'll need it to connect in a moment. It's also a good idea to note your Mac's IP address in case using this address doesn't work - click the Apple menu, hold down Alt (Option on some keyboards), and click System Information. In the window that appears click the Network heading in the list at the left, and look alongside either Wi-Fi or Ethernet at the right depending on which your Mac is using.
On the Mac you'll need to note either your computer's name or it's IP address, as highlighted here
Connecting to your Mac or PC from an iPad or iPhone
On the iOS device open the VNC Viewer app and click the plus icon at the top right. In the dialog box that appears, type the address you discovered in the paragraph above, and in the name field type something memorable and identifiable such as Mac Desktop.
Click Done, and then the Connect button. You'll be warned you're using an unencrypted connection. This is unfortunate but there's no way to avoid it, so tap the Connect link at the top right of the window. You'll then be prompted for the username and password of the Mac account you want to log into. You should type the "short" version of your username - usually your first name, or the first whole word of the longer version of the username.
You'll connect immediately and see a tips panel showing some control tricks. However, in short, the mouse cursor is represented by a small dot and you "shove" it around - push up on the screen, for example, and wherever the mouse cursor happens to be it will also move up. It can take a bit of getting used to. Tapping on the screen is the equivalent of clicking. Use the pinch-expand gesture to zoom in and out of the desktop.
To make a keyboard appear for typing, tap the keyboard icon on the toolbar at the top of the screen. To disconnect from the remote Mac, tap the X icon on the toolbar.
Using the free VNC Viewer app you can remotely control your Mac's desktop
If you want to connect to a Linux desktop you can install a VNC server package (just check your distro's package archive), and the same VNC Viewer app can be used to connect as described above.
Connect to remote Windows computers via your iPad or iPhone
Microsoft has a surprising number of apps available for iOS and one of them is Microsoft Remote Desktop. This lets you remotely connect to the desktop of Windows computers. You'll find it in the App Store and it's free of charge.
In typical Microsoft fashion not all versions of Windows have Remote Desktop compatibility built-in, and it's limited to the Ultimate, Business and Professional editions of XP, 7 and 8. (Although there are hacks to make it work on other versions - just Google.)
Before using Remote Desktop you'll need to ensure it's activated on the Windows computer you want to access. This can be done by clicking Start and typing Allow Remote Access To Your Computer. Then click the entry that appears in the results, and click Allow Connections Only From Computers Running Remote Desktop with Network Level Authentication (More Secure). Then click OK.
You'll need to activate Remote Desktop on Windows but it's only available on Ultimate, Business and Professional editions of the OS
You'll also need to know the IP address of the Windows computer. Click Start, then type cmd. In the DOS box that appears, type ipconfig and hit Enter. In the output, look for the line that reads IPv4 Address and make a note. Then close the DOS box.
Start the Microsoft Remote Desktop app on your iOS device and click the plus button in the top right, then select Add PC or Server. In the PC Name field, type the IP address you noted earlier. Tap the User Name field, then tap Add User Account. Now type the username and password of the Windows computer you want to access, and tap Save, and then Save in the parent dialog box. Then tap the icon for your new connection. You'll be asked immediately if you want to accept the security certificate. Tap the switch alongside Don't Ask Me Again and then tap Accept.
Connecting to a remote Windows desktop can be done using the free Microsoft Remote Desktop app
You'll see the remote PC's desktop and your fingertip moves the mouse cursor. To switch to touch controls, as if using a touchscreen PC, tap the IP address toolbar at the top of the screen, and select the option at the bottom right on the sidebars that appear. To bring up a keyboard for typing, tap the keyboard icon in the toolbar at the top of the screen.
To disconnect, tap the toolbar and then tap the X at the left of the thumbnail listing at the left of the screen.