Topological graph theory discusses, in most cases, graphs embedded in the plane (or other surfaces). For example, such plane graphs are sometimes regarded as the simplest town maps. Now, we consider a town having some pedestrian bridges, which cannot be realized by a plane graph. Its underlying graph can actually be regarded as a 1-plane graph. The notion of 1-plane and 1-planar graphs was first introduced by Ringel in connection with the problem of simultaneous coloring of the vertices and faces of plane graphs. In particular, in contrast to planarity testing, testing 1-planarity of a given graph is an NP-complete problem. Even though 1-planar graphs have been widely studied recently, we still know relatively little about them. In this chapter, we begin with formally defining 1-plane and 1-planar graphs and mainly focus on “maximal”, “maximum,” and “optimal” 1-planar graphs, which are relatively easy to treat. This chapter reviews some basic properties of these graphs.