Dust growth in protoplanetary discs ranges from micrometres to more than 12 orders of magnitude larger in size, which is required for planet formation. However, there are theoretical and experimental constraints to this process. On the one hand, the radial drift of dust towards the star is driven by aerodynamic drag between gas and dust, which is greatest for millimetre to centimetre grains and results in rapid accretion of such particles onto the star. The relative velocity between grains, on the other hand, which permits them to collide and grow, becomes the largest for these grains resulting in bouncing or fragmentation. Different solutions have been developed to overcome these obstacles, however they usually require specific operating circumstances. The main focus of this thesis was to develop our 3D Smoothed Particle Hydrodynamic (SPH) code to include grain growth and study the initial stages of planet formation in protoplanetary discs. To investigate that, I have modelled several systems with various physical initial conditions. Our simulation showed the formation of the so-called dust traps when we included fragmentation in the dust growth process due to back reaction from dust onto the gas. I also looked at multiple dust species and showed that dust clumps could form as a result of differences in dust migration velocities between various dust sizes. Finally, we have studied dust growth in discs involving dead zones and found that multiple density concentrations could form due to the combined effect of dead zone and fragmentation.