Massive multi-user (MU) multiple-input multiple-output (MIMO) will be a core technology in fifth-generation (5G) wireless systems as it offers significant improvements in spectral efficiency compared to existing multi-antenna technologies. The presence of hundreds of antenna elements at the base station (BS), however, results in excessively high hardware costs and power consumption, and requires high interconnect throughput between the baseband-processing unit and the radio unit. Massive MU-MIMO that uses low-resolution analog-to-digital and digital-to-analog converters (DACs) has the potential to address all these issues. In this paper, we focus on downlink precoding for massive MU-MIMO systems with 1-bit DACs at the BS. The objective is to design precoders that simultaneously mitigate MU interference and quantization artifacts. We propose two nonlinear 1-bit precoding algorithms and corresponding very large-scale integration (VLSI) designs. Our algorithms rely on biconvex relaxation, which enables the design of efficient 1-bit precoding algorithms that achieve superior error-rate performance compared with that of linear precoding algorithms followed by quantization. To showcase the efficacy of our algorithms, we design VLSI architectures that enable efficient 1-bit precoding for massive MU-MIMO systems, in which hundreds of antennas serve tens of user equipments. We present corresponding field-programmable gate array (FPGA) reference implementations to demonstrate that 1-bit precoding enables reliable and high-rate downlink data transmission in practical systems.