  We extend to finite elasticity the Data-Driven formulation of geometrically
linear elasticity presented in Conti, M\"uller, Ortiz, Arch.\ Ration.\ Mech.\
Anal.\ 229, 79-123, 2018. The main focus of this paper concerns the formulation
of a suitable framework in which the Data-Driven problem of finite elasticity
is well-posed in the sense of existence of solutions. We confine attention to
deformation gradients $F \in L^p(\Omega;\mathbb{R}^{n\times n})$ and first
Piola-Kirchhoff stresses $P \in L^q(\Omega;\mathbb{R}^{n\times n})$, with
$(p,q)\in(1,\infty)$ and $1/p+1/q=1$. We assume that the material behavior is
described by means of a material data set containing all the states $(F,P)$
that can be attained by the material, and develop germane notions of coercivity
and closedness of the material data set. Within this framework, we put forth
conditions ensuring the existence of solutions. We exhibit specific examples of
two- and three-dimensional material data sets that fit the present setting and
are compatible with material frame indifference.
