Non-archimedean analytic geometry nowadays comes in different flavors, including Tate's rigid-analytic geometry and Berkovich's theory of analytic spaces. From a category- theoretical point of view, these theories are essentially equivalent. In my talk, I will present yet another version of non-archimedean analytic geometry, a version which is category-theoretically non-equivalent to the above: uniformly rigid geometry. In a nutshell, a uniformly rigid space over a complete discretely valued field is a rigid space over that field together with an additional igidifying structure; this eponymous "uniform" structure is encoded in a coarser Grothendieck topology and a smaller sheaf of analytic functions. Informally speaking, uniformly rigid geometry makes it possible to systematically handle bounded functions on open non-archimedean discs. Uniformly rigid spaces naturally arise as generic fibers of formal schemes of formally finite type over the valuation ring of the base field. In my talk, I will define uniformly rigid spaces, I will give examples, and I will highlight the difference between uniformly and classical rigid geometry. I will also indicate potential applications of uniformly rigid geometry.