A multiple marker level-set (MLS) method is introduced for Direct Numerical Simulation of bubbles and droplets, which is integrated in a finite-volume framework on collocated unstructured grids of arbitrary element type. The MLS method is based on the idea of describing separate interfaces in the disperse phase with different levelset functions to prevent numerical and potentially unphysical coalescence of bubbles or droplets without excessive refinement. Thus, bubbles or droplets are able to approach each other closely, within the size of one grid cell, and can even collide, while artificial merging of the interfaces is prevented. The accuracy of the computational method implemented is examined for the problem of a single bubble rising in quiescent liquid. For the validation of the multiple marker level-set method we study the gravity-driven impact of a droplet on a liquid-liquid interface and finally it is applied on simulation of gravity-driven bubbly flow in a vertical pipe.