How does a domain transfer affect my website?

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People will still be able to access your site when it is being transferred, yes. Everything (such as Google Apps) will continue to work as long as your DNS servers (your nameservers) are set correctly on the domain name. If your nameservers are pointing to your registrar (GoDaddy) rather then your web host, then DNS services may well stop working when the domain is transferred out. The transfer time completely depends on the domain extension, but usually around 5 business days as long as you approve everything immediately via e-mail.