Yes. With the Shiny framework, you can create radically different looks and feels for your user-interfaces.
The early Shiny apps had side panels on the left and plots on the right. Developers could create highly customized HTML based user-interfaces from scratch in the Shiny framework, but required some additional knowledge.
After Shiny 0.9, the Shiny framework made it easier to use a variety of layouts. For examples see our gallery.