  We introduce a new class of states for bosonic quantum fields which extend
tensor network states to the continuum and generalize continuous matrix product
states (cMPS) to spatial dimensions $d\geq 2$. By construction, they are
Euclidean invariant, and are genuine continuum limits of discrete tensor
network states. Admitting both a functional integral and an operator
representation, they share the important properties of their discrete
counterparts: expressiveness, invariance under gauge transformations, simple
rescaling flow, and compact expressions for the $N$-point functions of local
observables. While we discuss mostly the continuous tensor network states
extending Projected Entangled Pair States (PEPS), we propose a generalization
bearing similarities with the continuum Multi-scale Entanglement
Renormalization Ansatz (cMERA).
