One would not expect to be able to conclude much about a link or a spatial graph by looking at a single projection. Yet in 1984, Menasco proved that if $G$ is a reduced, alternating, connected projection of a link $L$, then there is a sphere meeting $L$ in two points splitting the link into two non-trivial pieces if and only if there is a circle meeting $G$ in two points splitting the projection into two non-trivial pieces. Then in 1987, Kauffman, Murasugi, and Thistlethwaite proved Tait’s Conjecture of more than a century that any reduced, alternating projection of a link has a minimal number of crossings. Since the 80’s, these two important results have been generalized to other classes of links, tangles, and spatial graphs. In this talk we review known results, present counterexamples to some prior results about spatial graphs, and present new results for spatial graphs.