We study aggregation in turbulent flow by means of particle-resolved, direct numerical simulations. Mono-sized spheres with an attractive square-well potential are released in homogeneous, isotropic turbulence generated through linear forcing. Typical cases have a solids volume fraction of 0.08 and a ratio of the Kolmogorov scale over the primary sphere radius of O(0.1). The latter implies that the flow around the primary spheres is inhomogeneous. The simulations show the continuous formation and breakage of aggregates as a result of the turbulence and the attractive potential. The average size of the aggregates is a pronounced function of the strengths of turbulence and interaction potential. Fractal dimensions of the aggregates are in the range 1.4-1.8 for the cases studied.