I have been a great advocate of responsive web design and one of the reasons is the seamless brand experience you get across various devices, this just rocks. Thewebcitizen.com has been running on responsive design since late January 2013 with a great increase in user engagement.
So, do all websites need to run on responsive design? As long as you decide to go mobile then this is the way to go, I believe that creating a separate mobile site looses the brand harmony you get on responsive. Although many argue that the cost to run on responsive design is high, this will eventually go down, so, if you plan a website redesign go responsive.
Who does it great? Here are 5 examples of great responsive design.
From Mike Dickman: It is important to recognize that a responsive design is much easier to implement at the beginning of a web design rather than trying to create it once the site has been built. Make sure you have specific requirements for what shows up in your responsive design. What are the most important elements, or functions, of your website for a user to be able to navigate on a mobile device. Take your time and possibly consider different designs for phones and tablets.