  This is the first of a series of papers about \emph{quantization} in the
context of \emph{derived algebraic geometry}. In this first part, we introduce
the notion of \emph{$n$-shifted symplectic structures}, a generalization of the
notion of symplectic structures on smooth varieties and schemes, meaningful in
the setting of derived Artin n-stacks. We prove that classifying stacks of
reductive groups, as well as the derived stack of perfect complexes, carry
canonical 2-shifted symplectic structures. Our main existence theorem states
that for any derived Artin stack $F$ equipped with an $n$-shifted symplectic
structure, the derived mapping stack $\textbf{Map}(X,F)$ is equipped with a
canonical $(n-d)$-shifted symplectic structure as soon a $X$ satisfies a
Calabi-Yau condition in dimension $d$. These two results imply the existence of
many examples of derived moduli stacks equipped with $n$-shifted symplectic
structures, such as the derived moduli of perfect complexes on Calabi-Yau
varieties, or the derived moduli stack of perfect complexes of local systems on
a compact and oriented topological manifold. We also show that Lagrangian
intersections carry canonical (-1)-shifted symplectic structures.
