This cumulative thesis studies the topological transport of colloidal particles subject to an external magnetic field that changes its direction along a loop. The paramagnetic colloidal particles respond by moving on top of periodic magnetic patterns. I investigate five physical questions for various different situations using Brownian Dynamics simulations. The simulations are corroborated by experiments from other members in my group. I show how to build an adiabatic topologically protected discrete time crystal with three colloidal particles that interact via dipolar interactions and reside on a flower shaped periodic pattern. I extend this motive to various space time crystallines structures. I discuss the difference between a topologically protected and a geometric form of transport. I show how the transport changes from topological toward geometrical as a function of the particle number in an ensemble of several colloidal particles per unit cell. Dipolar interactions enforce the assembly of $n$ colloidal particles into a biped. I developed a topological protected polyglot programming of the biped walking on square patterns that lets bipeds of different length simultaneously walk into different predefined directions. I analyze the walking on square and hexagonal patterns with the center of mass gauge and the instantaneous center of rotation gauge, which allows me to decompose the motion into passive and active components. Finally at non-adiabatic speeds of driving I show how hydrodynamic interaction suppresses passive motion, stalls single colloidal particles unless still active bipeds assist their passive motion.
