Stability. I interpret this in a very general sense. If A implies B, does a small perturbation of A implies a small perturbation of B? This "theme" is omnipresent. I omit the discussion of the classic notion of Lyapunov's stability...

And I give only two examples, as required.

I. A dynamical system is called "structurally stable" or "robust", if a small perturbation of dynamics (in an appropriate function space) leads to the "same behavior", for example the preturbed system is topologically conjugate to the unperturbed one.

See, for example, MR0925417 Andronov, A. A.; Vitt, A. A.; Khaĭkin, S. È. Theory of oscillators, Dover Publications, Inc., New York, 1987.

For a more recent example of the same, see MR0732343 Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217.

II. Liouville's theorem says that conformal maps in Euclidean spaces of dimension $n\geq 3$ are "trivial", that is they are restrictions of M"obius transformations. What if a map is "close to conformal"? There is a nice precise definition of this: quasiconformal with small dilatation.
See the wonderful book of Reshetnyak,

MR1326375 Reshetnyak, Yu. G. Stability theorems in geometry and analysis, Kluwer Academic Publishers Group, Dordrecht, 1994.

III. An example of unsolved problem (due to Fedya Nazarov). A classical theorem of Rado says that if $f$ is a continuous function in a region in the complex plane, and $f$ is analytic on the set $\{ z:f(z)\neq 0\}$ then $f$ is analytic everywhere. What if $f$ is known to be analytic on the set $\{ z:|f(z)|<\epsilon\}$. Is it globally close to an analytic function in some sense? Give a quantitative estimate in terms of $\epsilon$.

Everyone can add her favorite examples of stability.

Stability. I interpret this in a very general sense. If A implies B, does a small perturbation of A implies a small perturbation of B? This "theme" is omnipresent. I omit the discussion of the classical notion of Lyapunov's stability...

And I give only two examples, as required.

I. A dynamical system is called "structurally stable" or "robust", if a small perturbation of dynamics (in an appropriate function space) leads to the "same behavior", for example the preturbed system is topologically conjugate to the unperturbed one.

See, for example, MR0925417 Andronov, A. A.; Vitt, A. A.; Khaĭkin, S. È. Theory of oscillators, Dover Publications, Inc., New York, 1987.

For a more recent example of the same, see MR0732343 Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217.

II. Liouville's theorem says that conformal maps in Euclidean spaces of dimension $n\geq 3$ are "trivial", that is they are restrictions of M"obius transformations. What if a map is "close to conformal"? There is a nice precise definition of this: quasiconformal with small dilatation.
See the wonderful book of Reshetnyak,

MR1326375 Reshetnyak, Yu. G. Stability theorems in geometry and analysis, Kluwer Academic Publishers Group, Dordrecht, 1994.

III. An example of unsolved problem (due to Fedya Nazarov). A classical theorem of Rado says that if $f$ is a continuous function in a region in the complex plane, and $f$ is analytic on the set $\{ z:f(z)\neq 0\}$ then $f$ is analytic everywhere. What if $f$ is known to be analytic on the set $\{ z:|f(z)|>\epsilon\}$. Is it globally close to an analytic function in some sense? Give a quantitative estimate in terms of $\epsilon$.

Everyone can add her favorite examples of stability.