When does a matrix equation have a solution?

Given a matrix equation Ax = b where A is a matrix and b is a column vector, what is a condition that would ensure that there is a column vector x that satisfies the equation?

Assume the dimensions are sensible, and feel free to provide multiple conditions.

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I assume you're looking for a condition considerably weaker than "A is invertible". –  Anton Geraschenko Oct 3 '09 at 14:49