shinyapps.io has a nice pattern for deploying applications and has a free tier, but it is a hosted service and not on-premise. So it all depends on what you are shooting to do.
If you're in an enterprise environment, RStudio Connect has a nice pattern for deploying applications as well (this is a paid professional product designed for enterprise teams). Otherwise, as @andresrcs mentioned, Shiny Server Open Source can be a nice fit for hosting Shiny applications in an open source paradigm. In this approach, you have to do the work he described yourself of reproducing your R version, packages, etc.