The　lack　of　experimental　investigations　on　graphene　fostered　researchers　to　focus　on　its　mechanical　modeling.　Being　graphene　a　one-atom-thick　sheet,　many　authors　developed　continuum　membrane　models　to　analyze　its　mechanical　behavior.　However,　an　entirely　nonlinear　approach　in　finite　elasticity　has　not　been　presented　so　far.　In　this　work,　the　equilibrium　problem　of　anisotropic　hyperelastic　graphene　membranes　is　addressed.　Strain　and　stress　measures　are　expressed　under　the　hypothesis　of　homogeneous　deformations　and　the　boundary-value　problem　is　formulated　for　a　graphene　membrane　subjected　to　biaxial　loads.　The　stability　of　the　equilibrium　configurations　is　assessed　through　an　energy　criterion.　Explicit　relations　between　stretches　and　stresses　of　the　membrane　are　derived　for　the　cases　of　uniaxial　and　equibiaxial　loads.　Unexpectedly,　bifurcation　and　multiple　equilibrium　solutions　are　found　when　graphene　is　subjected　to　equibiaxial　loads.　A　linearization　of　the　finite　theory　is　presented　and　the　expressions　of　Young’s　modulus　and　Poisson’s　ratio　of　graphene　are　derived.　The　formulation　proposed　in　this　work　may　be　the　basis　for　accurate　investigations　of　the　mechanics　of　graphene　subjected　to　large　deformations.　©　2021,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.