First, if your existing concrete is no longer attractive, you have options to improve the appearance without ripping it out. Specially formulated concrete paints and stains exist that you can use to cover discolor concrete. Most paint stores as well as Home Depot carry it. It's an easy and inexpensive solutions.
Second, in some cases you can also cover it with slate tiles and specially formulated grouts and adhesives as I spoke about some time ago in this article.
Finally, there are even more elaborate, albeit more costly, options that turn concrete from the standard stuff of driveways and sidewalks to beautiful accents for basement and outdoors. The picture above is not brick but a concrete veneer and stain technique. Also possible are staining, coloring, etching, stamping and more. Take a look at this slideshow. You won't believe that some of this is concrete.