Abstract: In this paper the δ-shock front problem is studied. For some classes of hyperbolic systems of conservation laws (in several space dimension, too) we introduce the definitions of a δ-shock wave type solution relevant to the front problem. The Rankine–Hugoniot conditions for δ-shocks are analyzed from both geometrical and physical points of view. δ-Shock balance relations connected with area and mass transportation are derived. The geometric aspect of δ-shock formation from sufficiently smooth compactly supported initial data is considered. We study the propagation of δ-shocks in two hyperbolic systems of conservation laws. In the one-dimensional case, we consider the system

u_t+(f(u)-v)_x=0,    v_t+(g(u))_x=0,

where f(u) and g(u) are polynomials of degree n and n+1, respectively, n is even. The well-known Keyfitz–Kranzer system

u_t+(u^2-v)_x=0,    v_t+(u^3/3-u)_x=0

is a particular case of the last system. In the multidimensional case a non-conservative form of zero-pressure gas dynamics system

ρ_t+∇·(ρU)=0,    U_t+(U·∇)U=0,

is studied. This system has been used to describe the formation of large-scale structures of the universe. Both systems have several “bad” properties (see below). As far as we know, δ-shock wave type solutions for them have never been constructed.