Sufficient conditions for $\lim_{m \to \infty}\lim_{n \to \infty} a_{mn} = \lim_{m \to \infty}\lim_{n \to \infty} a_{mn}$

Question

Recently I came across with some problems concerning the following fundamental problem in the advanced calculus: $$\lim_{m \to \infty}\lim_{n \to \infty} a_{mn} = \lim_{m \to \infty}\lim_{n \to \infty} a_{mn}$$

Is there any sufficient(or necessary and sufficient, if there is) conditions for this limit equality? I know from the Fubini's theorem that $\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}a_{mn}=\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}a_{mn}$ under some conditions for $a_{mn}$