The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show they are equal in some good domains.

Also, one can study something like moduli spaces by deforming a point to formal neighborhood, and use this (plus obstruction theory) to detect dimension and other properties.

Therefore, the idea of deformation is very common and useful , and there are many famous applications:

- Deligne proves Hodge implies absolute Hodge for abelian varieties by deforming the general case to CM abelian varieties (using principle $B$).
- If $X,Y$ are two smooth projective curves of genus $>1$, then $Hom_{nonconst}(X,Y)$ is finite by studying the Hom scheme using deformation theory.
- We can use deformation to detect dimension of moduli spaces we hope to understand, or prove some smoothness/non-smoothness theorems ($\mathcal{M_g}$, $Bun_G$, Hilbert schemes, smoothness of Jacobian for smooth projective curves, modulis of Calabi-Yau by Tian–Todorov).
- Lifting theorems by considering infinitesimal thickenings and algebraic theorems, for example Grothendieck for curves, Serre-Tate for ordinary abelian varieties, Deligne for K3 surfaces (up to a finite extension).
- Generic vanishing theorem for abelian varieties.
- Perverse continuation principle.
- Mori's bound and break techniques.
- Deformation of Galois representations.
- Gross’s proof of Chowla-Selberg formula.
- Tits deformation principle.
- Deformation to the normal cone.
- The proof of standard conjectures for any smooth projective variety of $K3^{[n]}$ type (see https://arxiv.org/abs/1009.0413).

There is of course more, what are some other applications of the idea of deformation in algebraic geometry, number theory and representation theory etc? I believe some are still very important but maybe non-experts don't know those examples and ideas.

In some sense, proving something in good case and using density argument may be regarded as applying deformation as well. As a very naive example, one can prove Cayley-Hamilton theorem by first considering diagonalizable matrices.