  For germs of holomorphic functions $f : (\mathbf{C}^{m+1},0) \to
(\mathbf{C},0)$, $g : (\mathbf{C}^{n+1},0) \to (\mathbf{C},0)$ having an
isolated critical point at 0 with value 0, the classical Thom-Sebastiani
theorem describes the vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (and
its monodromy) as a tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where $(f
\oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. We prove
algebraic variants and generalizations of this result in \'etale cohomology
over fields of any characteristic, where the tensor product is replaced by a
certain local convolution product, as suggested by Deligne. They generalize
arXiv:1105.5210. The main ingredient is a K\"unneth formula for $R\Psi$ in the
framework of Deligne's theory of nearby cycles over general bases. In the last
section, we study the tame case, and the relations between tensor and
convolution products, in both global and local situations.
