  In this review we consider the quadratic Metric-Affine Gauge gravity
Lagrangian, which contains all the algebraic invariants up to quadratic order
in torsion, nonmetricity and curvature. The goal will be to collect the known
exact solutions for this theory that describe the propagation of a
gravitational wave and some useful properties. We concentrate on solutions
whose connections are different from the Levi-Civita one and provide also the
extension of already known axial torsion solutions to more general Lagrangians.
At the end, we will briefly discuss colliding waves and Riemannian solutions.
