Well, then I'll start with the most obvious generalized solutions:

• weak solutions to PDEs
• Schwartz's generalized Functions aka Distributions,
• Colombeau's algebra(s) of generalized functions and
• various other kinds of generalized functions
• Quasi-Minima in functional analysis: A quasi-minimum of a functional $\mathcal{F}$ is a $u$ such that $\mathcal{F}u\leq Q\mathcal{F}v$ for all $v$ (with some constant $Q\geq 1$)
• Every solution of an polynomial equation within $\mathbb{C}$ can be a generalized solution if you're problem is something that has only real (maybe some geometric problem) or only integer or even only natural (maybe something from number theory) solutions. But considering all complex solution to your particular equation often gives a very elegant treatment of the problem.