We overview the ways in which electric fields can be used for on-chip manipulation and assembly of colloidal particles. Particles suspended in water readily respond to alternating (AC) or direct current (DC) electric fields. Charged particles in DC fields are moved towards oppositely charged electrodes by electrophoresis. Dielectrophoresis, particle mobility in AC fields, allows precise manipulation of particles through a range of parameters including field strength and frequency and electrode geometry. Simultaneously, DC or AC electrokinetics may drive liquid flows inside the experimental cells, which also leads to transport and redistribution of the suspended particles. Examples of dielectrophoretic manipulation and assembly of nanoparticles and microparticles by planar on-chip electrodes are presented. The structures assembled include conductive microwires from metallic nanoparticles and switchable two-dimensional crystals from polymer microspheres. We also discuss how dielectrophoresis and AC electrokinetics can be used in droplet-based microfluidic chips, biosensors, and devices for collection of particles from diluted suspensions.