An important theorem by Beilinson describes the bounded derived category of coherent sheaves on $mathbb{P}^n$, yielding in particular a resolution of every coherent sheaf on $mathbb{P}^n$ in terms of the vector bundles $Omega_{mathbb{P}^n}^j(j)$ for $0le jle n$. This theorem is here extended to weighted projective spaces. To this purpose we consider, instead of the usual category of coherent sheaves on $mathbb{P}({rm w})$ (the weighted projective space of weights $rm w=({rm w}_0,dots,{rm w}_n)$), a suitable category of graded coherent sheaves (the two categories are equivalent if and only if ${rm w}_0=cdots={rm w}_n=1$, i.e. $mathbb{P}({rm w})= mathbb{P}^n$), obtained by endowing $mathbb{P}({rm w})$ with a natural graded structure sheaf. The resulting graded ringed space $overline{mathbb{P}}({rm w})$ is an example of graded scheme (in chapter 1 graded schemes are defined and studied in some greater generality than is needed in the rest of the work).Then in chapter 2 we prove for graded coherent sheaves on $overline{mathbb{P}}({rm w})$ a result which is very similar to Beilinson's theorem on $mathbb{P}^n$, with the main difference that the resolution involves, besides $Omega_{overline{mathbb{P}}({rm w})}^j(j)$ for $0le jle n$, also $mathcal{O}_{overline{mathbb{P}}({rm w})}(1)$ for $n-sum_{i=0}^n{rm w}_i1$. This weighted version of Beilinson's theorem is then applied in chapter 3 to prove a structure theorem for good birational weighted canonical projections of surfaces of general type (i.e., for morphisms, which are birational onto the image, from a minimal surface of general type $S$ into a $3$-dimensional $mathbb{P}({rm w})$, induced by $4$ sections $sigma_iin H(S, mathcal{O}_S({rm w}_iK_S))$).This is a generalization of a theorem by Catanese and Schreyer (who treated the case of projections into $mathbb{P}^3$), and is mainly interesting for irregular surfaces, since in the regular case a similar but simpler result (due to Catanese) was already known. The theorem essentially states that giving a good birational weighted canonical projection is equivalent to giving a symmetric morphism of (graded) vector bundles on $overline{mathbb{P}}({rm w})$, satisfying some suitable conditions. Such a morphism is then explicitly determined in chapter 4 for a family of surfaces with numerical invariants $p_g=q=2$, $K^2=4$, projected into $mathbb{P}(1,1,2,3)$.