Higher analogues of algebraic and geometric structures studied in symplectic geometry naturally arise on manifolds equipped with a closed non-degenerate form of degree $> 2$. Traditionally, these ``multisymplectic´´ manifolds have been used to describe classical field theories. In this talk, I will first explain how a multisymplectic manifold gives an $L_{\infty}$-algebra of ``Hamiltonian´´ differential forms, just as a symplectic manifold gives a Poisson algebra of functions. I will then discuss the (pre)quantization of such manifolds. In the symplectic case, the relevant structures include principal $U(1)$-bundles and their corresponding Atiyah Lie algebroids. Similarly, for the degree 3 case, I will show that the relevant structures include $U(1)$-gerbes and their corresponding ``exact Courant algebroids´´. A gerbe is a kind of stack which can be understood as the higher analogue of a principal bundle, while an exact Courant algebroid is a vector bundle which plays the role of a higher Atiyah algebroid. The space of global sections of the Courant algebroid is an $L_{\infty}$-algebra, which can be used to give a quantization of the $L_{\infty}$-algebra of Hamiltonian forms.