(One computation, three answers)
\(K_{\PP^n} = -(n+1)H\), so \(-K_{\PP^n} = \OO(n+1)\) is very ample and \(\PP^n\) is Fano. For \(D \subseteq \PP^n\) a smooth hypersurface of degree \(d\), adjunction gives \[ K_D = \restrictionof{ (K_{\PP^n} + D) }{D} = \restrictionof{ \qty{ -(n+1)H + dH } }{D} = (d - n - 1) \restrictionof{H}{D} . \] So \(D\) is Fano for \(d < n+1\), Calabi–Yau for \(d = n+1\), and of general type for \(d > n+1\).
This computation supplies examples in either class on request, and a single adjunction line covers both. The quadric surface in \(\PP^3\) is \(d = 2 < 4\), hence del Pezzo, and indeed \(\PP^1 \times \PP^1\); the quartic surface in \(\PP^3\) is \(d = 4 = n+1\), the K3 surface; the quintic threefold in \(\PP^4\) is the Calabi–Yau of mirror symmetry. For \(n = 2\) the same computation is the plane-curve genus formula \(g = \binom{d-1}{2}\), with \(d \leq 2\) rational, \(d = 3\) elliptic, \(d \geq 4\) general type.
The vanishing \(h^j(\OO_D) = 0\) demanded by the strict Calabi–Yau convention is not part of the adjunction computation; it comes from Lefschetz, which says the middle-dimension cohomology is the only new one.
Dependencies
D-VARFANOFano, Calabi--Yau, and general typeuses