  We provide existence, uniqueness and stability results for affine stochastic
Volterra equations with $L^1$-kernels and jumps. Such equations arise as
scaling limits of branching processes in population genetics and self-exciting
Hawkes processes in mathematical finance. The strategy we adopt for the
existence part is based on approximations using stochastic Volterra equations
with $L^2$-kernels combined with a general stability result. Most importantly,
we establish weak uniqueness using a duality argument on the Fourier--Laplace
transform via a deterministic Riccati--Volterra integral equation. We
illustrate the applicability of our results on Hawkes processes and a class of
hyper-rough Volterra Heston models with a Hurst index $H \in (-1/2,1/2]$.
