  The pro-\'etale fundamental group of a scheme, introduced by Bhatt and
Scholze, generalizes formerly known fundamental groups -- the usual \'etale
fundamental group $\pi_1^{\mathrm{et}}$ defined in SGA1 and the more general
group defined in SGA3. It controls local systems in the pro-\'etale topology
and leads to an interesting class of "geometric covers" of schemes,
generalizing finite \'etale covers.
  We prove the homotopy exact sequence over a field for the pro-\'etale
fundamental group of a geometrically connected scheme $X$ of finite type over a
field $k$, i.e. that the sequence $$1 \rightarrow
\pi_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{proet}}(X)
\rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the
map $\pi_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{proet}}(X)$
is a topological embedding. On the way, we prove a general van Kampen theorem
and the K\"unneth formula for the pro-\'etale fundamental group.
