Is there a proof that any graph is "drawable" on a 2D surface?

Question

Are there any theorems that say something formal about the fact that any graph is drawable on a 2D surface, and can be mapped to a 2D array of pixels if the pixels are infinitely small?

EDIT: No restrictions on whether edges may cross.

Answer 1

Of course any graph is drawable on a plane - pick $n$ arbitrary distinct points on the plane to be the nodes, and draw line segments between them for the edges. Of course, this drawing might be quite ugly, especially if you don't choose "good" points. But it's still a drawing.

Kuratowski's theorem gives conditions under which it's possible to draw a graph in the plane without crossing the edges.

Fary's theorem extends this by saying that as long as the graph is simple (at most one edge between any two nodes, and no nodes with an edge going to themselves), you can actually get a crossing-free drawing with straight lines for the edges.

There is also an entire area of computer science dedicated to finding algorithms for finding pretty drawings of graphs.