  Laser-plasma acceleration is an emerging technique for accelerating electrons
to high energies over very short distances. The accelerated electron bunches
have femtosecond duration, making them particularly relevant for applications
such as ultrafast imaging or femtosecond X-ray generation. Current laser-plasma
accelerators are typically driven by Joule-class laser systems that have two
main drawbacks: their relatively large scale and their low repetition-rate,
with a few shots per second at best. The accelerated electron beams have
energies ranging from 100 MeV to multi-GeV, however a MeV electron source would
be more suited to many societal and scientific applications. Here, we
demonstrate a compact and reliable laser-plasma accelerator producing
high-quality few-MeV electron beams at kilohertz repetition rate. This
breakthrough was made possible by using near-single-cycle light pulses, which
lowered the required laser energy for driving the accelerator by three orders
of magnitude, thus enabling high repetition-rate operation and dramatic
downsizing of the laser system. The measured electron bunches are collimated,
with an energy distribution that peaks at 5 MeV and contains up to 1 pC of
charge. Numerical simulations reproduce all experimental features and indicate
that the electron bunches are only $\sim 1$ fs long. We anticipate that the
advent of these kHz femtosecond relativistic electron sources will pave the way
to wide-impact applications, such as ultrafast electron diffraction in
materials with an unprecedented sub-10 fs resolution.
