  We study existence, regularity, and qualitative properties of solutions to
linear problems involving higher-order fractional Laplacians $(-\Delta)^s$ for
any $s>1$. Using the nonlocal properties of these operators, we provide an
explicit counterexample to general maximum principles for $s\in(n,n+1)$ with
$n\in\mathbb N$ odd; moreover, using a representation formula for solutions, we
derive regularity and positivity preserving properties whenever the domain is
the whole space or a ball. In the case of the whole space we analyze the Riesz
kernel, which provides a fundamental solution, while in the case of the ball we
show the validity of Boggio's representation formula for all integer and
fractional powers of the Laplacian $s>0$. Our proofs rely on characterizations
of $s$-harmonic functions using higher-order Martin kernels, on a decomposition
of Boggio's formula, and on elliptic regularity theory.
