We introduce Quasiconformal Neural Networks (QNNs), a novel framework that integrates quasiconformal maps into neural architectures, providing a rigorous mathematical basis for handling non-Euclidean data. QNNs control geometric distortions using bounded maximal dilatation across network layers, preserving essential data structures. We present theoretical results that guarantee the stability and geometric consistency of QNNs. This work opens new avenues in geometric deep learning, particularly for applications involving complex topologies, with significant implications for fields such as image registration and medical imaging.