In the last two decades, advances in synthetic, experimental and modeling/simulation methodologies have considerably enhanced our understanding of colloidal suspension rheology and put the field at the forefront of soft matter research. Recent accomplishments include the ability to tailor the flow of colloidal materials via controlled changes of particle microstructure and interactions. Whereas hard sphere suspensions have been the most widely studied colloidal system, there is no richer type of particles than soft colloids in this respect. Yet, despite the remarkable progress in the field, many outstanding challenges remain in our quest to link particle microstructure to macroscopic properties and eventually design appropriate soft composites. Addressing them will provide the route towards novel responsive systems with hierarchical structures and multiple functionalities. Here we discuss the key structural and rheological parameters which determine the tunable rheology of dense soft deformable colloids. We restrict our discussion to non-crystallizing suspensions of spherical particles without electrostatic or enthalpic interactions. Display omitted •Classification of soft colloids using particle elasticity and interaction potential•Softness impacts significantly vitrification, crystallization and jamming.•Rheology is an exquisite tool to probe thermodynamic and kinetic transitions.•Mixtures of soft colloids exhibit rich state diagrams with new glassy and gel states.•Design strategies for colloidal materials with desired flow properties