Given a smooth complete intersection $X=D_{1} \cap D_{2} \cap \cdots \cap D_{k} \subset \mathbb{P}^{n}$ with ${\rm deg}\; D_i=d_i$, one can easily show that $\omega_{X} \simeq \mathcal{O}_{X}(\sum_{i=1}^{k} d_{i} -n-1)$, using induction on the number of hypersurfaces and the usual conormal sequence.

Here is the question.

Suppose $X \subset \mathbb{P}^{n}$ is a smooth projective variety of degree $d$, not necessarily a complete intersection. How to understand $\omega_{X}$ in terms of the embedding? Is it even necessarily true that $\omega_{X}$ is restricted from a line bundle on $\mathbb{P}^{n}$?

Similarly, how to work out the cohomology of $\mathcal{O}_X$ and $\omega_X$? Does this only depend on the degree of $X$?