I will begin by recalling why and how one should add analysis to cyclic homology theories using bornologies. Then I speak about the definition of analytic cyclic homology theories using a class of analytically nilpotent algebras and universal extensions. This approach was introduced by Cuntz and Quillen and used by them to prove excision for periodic cyclic homology. It is ideal to study the analytic variants of cyclic cohomology, which lack a direct link to Hochschild cohomology. In the end, I will survey my recent work on analytic cyclic homology for algebras over the p-adic numbers and over finite fields.