WhatsApp had released it's web client 'WhatsApp Web' before sometime. It's initial version was only compatible with Chrome.
Good news for those people whose default browser is not Chrome is that WhatsApp web client is now compatible with Firefox and Opera.
Although, the WhatsApp Web feature is currently available for Windows, Android and Blackberry users only so the iOS and Safari users are still in a lurch.
The sign-up process is pretty much the same. You will be prompted to scan a QR code on the web client through your WhatsApp mobile app.
Here's the link to the Web Client: https://web.whatsapp.com/