  We consider billiard ball motion in a convex domain of the Euclidean plane
bounded by a piece-wise smooth curve influenced by the constant magnetic field.
We show that if there exists a polynomial in velocities integral of the
magnetic billiard flow then every smooth piece $\gamma$ of the boundary must be
algebraic and either is a circle or satisfies very strong restrictions. In
particular in the case of ellipse it follows that magnetic billiard is
algebraically not integrable for all magnitudes of the magnetic field. We
conjecture that circle is the only integrable magnetic billiard not only in the
algebraic sense, but for a broader meaning of integrability. We also introduce
the model of Outer magnetic billiards. As an application of our method we prove
analogous results on algebraically integrable Outer magnetic billiards.
