Victor, I don't understand your claim that $C^0$-semigroups aren't really semigroups. You are not free to decide for all the mathematical community what is a semigroup (I guess that you are interested only on discrete semigroups, aren't you ?).

$C^0$-semigroups are fundamental in PDEs (in probability too as mentioned by Steinhurst). The reason is that a lot of evolution PDEs (basically all parabolic ones, like the heat equation, or Navier-Stokes) can be solved only forward but not backward. In linear PDEs, this is a consequence of the Uniform Boundedness Principle (= Banach-Steinhaus Theorem). There is a nice theory relating operators and semigroups, the former being the generator of the latter. In the linear case, a fundamental result is the Hille-Yosida Theorem. Subsequent tools are Duhamel's principle and Trotter's formula. A part of the theory extends to nonlinear semigroups.

**Edit**. John B. expresses a doubt on the fundamental aspect of semigroups, compared with the evolution equations from which they arise. Let me say that semi-groups say much more, for the following reason. Evolutionary PDEs have classical solutions only when the initial data $u_0$ is smooth enough, typically when $u_0$ belongs to the so-called *domain* of the generator. This result can *never* be used to pass from a linear context to a non-linear one *via* the Duhamel's principle. In other words, in order to have a well-posed Cauchy-problem in Hadamard's sense, we need to invent a notion of weaker solutions ; this is where the semi-group theory comes into play.