Ultrasound directed self-assembly (DSA) relies on the acoustic radiation force associated with a standing ultrasound wave to organize particles dispersed in a fluid medium into specific patterns. State-of-the-art ultrasound DSA methods can only organize particles into (quasi-)periodic patterns, limited by the use of single-frequency ultrasound wave fields. Acoustic holography and acoustic waveguides provide alternatives to assembling complex patterns of particles, but generally provide low spatial accuracy and are not re-configurable because they require custom hardware for each specific pattern of particles, which is impractical. We introduce multi-frequency ultrasound wave fields to organize particles in non-periodic patterns. We theoretically derive and experimentally validate a solution methodology to determine the operating parameters (frequency, amplitude, phase) of any number and spatial arrangement of ultrasound transducers, required to assemble spherical particles dispersed in an inviscid fluid medium into any specific two-dimensional pattern. The results show that multi-frequency ultrasound DSA enables the assembly of complex, non-periodic patterns of particles with substantially fewer ultrasound transducers than single-frequency ultrasound DSA, and without incurring a penalty in terms of accuracy. The results of this work fundamentally transform the state-of-the-art knowledge of ultrasound DSA. Multi-frequency ultrasound wave fields enable a near-unlimited complexity of patterns of particles that can be assembled, increasing the relevance of the technology to practical implementation in engineering applications such as manufacturing of engineered composite materials that derive their properties from the spatial organization of the filler in the matrix material. Although this work focuses specifically on ultrasound wave fields, the theoretical model is valid for all wave phenomena. / Master of Science / Ultrasound directed self-assembly (DSA) is the process where particles dispersed in a fluid medium assemble into specific patterns due to their interactions with a sound wave and/or other particles. Current ultrasound DSA methods use a single-frequency ultrasound wave to assemble particles into specific patterns, which creates repeating patterns within the fluid medium. Other methods of assembling particles that allow for more complex, non-repeating patterns generally provide low spatial accuracy and do not allow dynamically changing the pattern as they require custom hardware for each specific pattern of particles, rendering these methods impractical. We use many ultrasound waves each with a different frequency to organize particles into complex, non-repeating patterns, which we call multi-frequency ultrasound DSA. We theoretically derive and experimentally validate a method that allows us to assemble any specific two-dimensional pattern of particles using multi-frequency ultrasound DSA. The results show that multi-frequency ultrasound DSA enables the assembly of complex, non-repeating patterns of particles with substantially resources than single-frequency ultrasound DSA, and without incurring a penalty in terms of accuracy. Multi-frequency ultrasound DSA enables a near-unlimited complexity of patterns of particles that can be assembled, increasing the relevance of the technology to practical implementation in engineering applications.