A group-presentation is said to be "balanced" if it has the same number of generators as relations. I shall begin by explaining why this is an interesting definition, with emphasis on its topological significance. I shall also present several explicit classes of examples. I want to present recent results concerning the complexity of the Andrews-Curtis conjecture which, as I shall explain, is related to the smooth 4-dimensional Poincare conjecture. I shall also sketch the role that balanced presentations play in the my work with Grunewald on Grothendieck's problem concerning representations of groups.