Using topological graph theory we will develop planar graph models to study the properties of special simple families of plane and spherical Jordan curves. Utilizing these procedures we solved some geometrical and topological problems. Among the others we answered some of the problems and conjectures of Professor Grunbaum. In this talk I will present some of our results on convex and strongly convex, simple, irreducible planar and spherical Venn diagrams. One of these results finally and fully corrects the erroneous statements that started with John Venn more than a century ago in 1880 and have been repeated frequently by others since then. We also will solve a conjecture of Grunbaum: Every Venn diagram on n curves can be extended to a Venn diagram of n+1 curves by the addition of a suitable simple closed Jordan curve. This finally solves a problem that goes back a century to John Venn's paper in 1880. We also will discuss a related conjecture of Peter Winkler. I will raise several problems and conjectures that arise from our work. I believe the talk will interest graph theorists and topologists as well as geometers, but it is accessible for any faculty or student with minimal knowledge of graph theory, topology, and geometry. Some of the results are joint results with Kiran B. Chilakamarri and/or Raymond E. Pippert (IPFW).