Is forward scattering = no scattering? What is forward scattering? If it is equivalent to no scattering, then why not call it "no scattering"? 
 A: If you want an even more everyday example than Emilio Pisanty's example: "no scattering" would mean that the would be scattering object in question (modelled by the short range potential in Emilio's answer)   would beget no change the the forward travelling wave. Otherwise put, an observer sensing the incoming plane wave could not tell whether or not the object were in between them and the source of (wontedly plane for these scenarios) waves. Forward scattering means that the scattering object leaves signs of its presence in the forward travelling wave. 
As an illustration, I don't know whether they still do this in Britain, but it used to be that one must by law buy a licence to have a television set. To enforce this law, the state had people driving around in vans (see them in action here) detecting unlicensed televisions from the scattered wave that must always arise from the television's antenna system whenever it interacts with the electromagnetic field. This scattered wave can be sensed at almost any orientation relative to the receiver and transmitter. In this scenario, "no scattering" would mean that you would be safe from the TV detector man.
A: The optical theorem (https://en.wikipedia.org/wiki/Optical_theorem) relates the imaginary part of the forward scattering amplitude to the total scattering cross-section,
$ \sigma_\mathrm{tot}=\frac{4\pi}{k}~\mathrm{Im}\,f(0)$, so if $ \sigma_\mathrm{tot}$ is non-zero, so is the forward scattering amplitude.
A: Forward scattering need not be equivalent to "no scattering" - and, indeed, will only rarely be indistinguishable from it.
In the usual scattering-theory setup, you have an electron coming in in a plane wave 
$$\psi(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}=e^{ikz}$$
and impinging on some short-range potential. This will add to the wavefunction a scattered wave
$$\psi_\text{scattered}(\mathbf{r})=F(\theta,\phi)\frac 1r e^{ikr}.$$
The form factor $F(\theta,\phi)$ governs the angular structure of the scattered wave, and the case where $\theta=0$ is called forward scattering.
Note that:


*

*The forward-scattered wave is part of a spherical wave and its amplitude decays with the distance from the scattering centre in a different way to the incoming wave. In practice, the incoming beam will also suffer from wavepacket spreading, but in general the forward-scattered wave will be weaker unless special scattering conditions are at play.

*The form factor in general includes a phase. This means that the forward-scattered wave will interfere nontrivially with the incoming beam, providing a delay in the phase of the final wave.
A: In QFT, forward scattering means $\theta\approx0$. So it is basically some kind of small angle approximation. This is the answer to my question. 
($\theta$ is the usual angle in spherical coordinates)
