We investigated the interaction of a laser-induced cavitation bubble with an elastic tissue model by high-speed photography with up to 5 Mill. frames/sec. The elastic material consisted of a transparent polyacrylamide (PAA) gel whose elastic properties can be controlled by modifying the water content to mimic various biological tissues. The elastic modulus E of the PAA sample was varied between 0.017 and 2 MPa. The dimensionless bubble-boundary distance γ(distance between laser focus and sample boundary, scaled by the maximum bubble radius) was for each value of E varied between γ = 0 and γ = 2.2. In this parameter space, we determined the jetting behavior, jet velocity, jet penetration into the PAA sample and bubble- induced removal of PAA material. The jetting behavior varies between unidirectional jets towards or away from the boundary, and formation of an annular jet which results in bubble splitting and subsequent formation of two very fast axial jets flowing simultaneously towards the boundary and away from it. General principles of the formation of annular and axial jets are discussed which allow to interpret the complex dynamics. The liquid jet directed away form the boundary reaches a maximum velocity between 300 m/s and 600 m/s (depending on E) while the peak velocity of the jet directed towards the boundary ranges between 400 m/s and 960 m/s. The peak velocities near an elastic material are 10 times higher than close to a rigid boundary. The liquid jet penetrates PAA samples with an elastic modulus in the intermediate range 0.12 < E < 0.4 MPa. In this same range of elastic moduli and for small γ-values, PAA material is ejected into the surrounding liquid due to the elastic rebound of the sample surface that was deformed during bubble expansion. The surface of the PAA sample is, furthermore, lifted during bubble collapse when a region of low pressure develops between bubble and sample. For stiffer boundaries, only an axial liquid jet towards the boundary is formed, similar to the bubble dynamics next to a rigid wall. For softer sample, the liquid jet is directed away from the boundary, and material is torn off the PAA sample during bubble collapse, if the bubble is produced close to the boundary. These processes play an important role for the efficiency and side effects of pulsed laser surgery inside the human body.