  Microscopic models for heavy-fermion materials often assume a local, i.e.,
momentum-independent, hybridization between the conduction band and the
local-moment f electrons. Motivated by recent experiments, we consider
situations where this neglect of momentum dependence is inappropriate, namely
when the hybridization function has nodes in momentum space. We explore the
thermodynamic and optical properties of the highly anisotropic heavy Fermi
liquid, resulting from Kondo screening in a higher angular-momentum channel.
The dichotomy in momentum space has interesting consequences: While e.g. the
low-temperature specific heat is dominated by heavy quasiparticles, the
electrical conductivity at intermediate temperatures is carried by unhybridized
light electrons. We then discuss aspects of the competition between Kondo
effect and ordering phenomena induced by inter-moment exchange: We propose that
the strong momentum-space anisotropy plays a vital role in selecting competing
phases. Explicit results are obtained for the interplay of unconventional
hybridization with unconventional, magnetically mediated, superconductivity,
utilizing variants of large-N mean-field theory. We make connections to recent
experiments on CeCoIn5 and other heavy-fermion materials.
