Abstract:  In usual Hida theory, one constructs a module over weight space whose specialitzation to sufficiently regular weight is a space over classical ordinary modular forms.  The goal of higher Hida theory is to construct a perfect complex of modules over (part of) weight space whose specialization to classical *non-regular* weights is a complex which (in favorable circumstances) computes the coherent cohomology of the assoicated Shimura variety in that weight.  In this talk, we outline a construction of these complexes for GSp(4) and certain families of irregular weights.