This paper presents quantum error-correcting codes that encode a finite-dimensional quantum system (a qudit) into an infinite-dimensional system, such as a harmonic oscillator. These codes protect against errors that shift the values of the canonical variables q (position) and p (momentum). The codes are based on the noncommutative geometry of phase space and can be used to implement fault-tolerant quantum computation and communication using harmonic oscillators or rotors. The codes are constructed by considering the large-d limit of shift-resistant codes for discrete systems, and they can be used to protect against small diffusive motions of all particles in a code block. The paper also discusses how these codes can be used to protect against amplitude damping and unitary errors, and how they can be combined with conventional finite-dimensional codes to provide protection against errors that heavily damage a subset of oscillators. The paper also describes how encoded quantum states can be processed fault-tolerantly using linear optical operations, squeezing, homodyne detection, and photon counting. The codes are shown to be effective in protecting against sufficiently weak diffusive phenomena that cause the position and momentum of an oscillator to drift, or against losses that cause the amplitude of an oscillator to decay. The paper also discusses the use of continuous-variable codes to establish lower bounds on the quantum capacity of Gaussian quantum channels. The codes are shown to be effective in protecting against errors that shift the values of the canonical variables q and p, and they can be used to implement fault-tolerant quantum computation and communication using harmonic oscillators or rotors. The paper also discusses the use of continuous-variable codes for many oscillators, and how they can be constructed from lattices in higher-dimensional phase space. The paper also discusses the use of concatenated codes to protect against a broader class of errors than small diffusive shifts applied to each oscillator. The paper concludes with a discussion of the physical realization of the coding schemes and the implications for quantum computing and communication.This paper presents quantum error-correcting codes that encode a finite-dimensional quantum system (a qudit) into an infinite-dimensional system, such as a harmonic oscillator. These codes protect against errors that shift the values of the canonical variables q (position) and p (momentum). The codes are based on the noncommutative geometry of phase space and can be used to implement fault-tolerant quantum computation and communication using harmonic oscillators or rotors. The codes are constructed by considering the large-d limit of shift-resistant codes for discrete systems, and they can be used to protect against small diffusive motions of all particles in a code block. The paper also discusses how these codes can be used to protect against amplitude damping and unitary errors, and how they can be combined with conventional finite-dimensional codes to provide protection against errors that heavily damage a subset of oscillators. The paper also describes how encoded quantum states can be processed fault-tolerantly using linear optical operations, squeezing, homodyne detection, and photon counting. The codes are shown to be effective in protecting against sufficiently weak diffusive phenomena that cause the position and momentum of an oscillator to drift, or against losses that cause the amplitude of an oscillator to decay. The paper also discusses the use of continuous-variable codes to establish lower bounds on the quantum capacity of Gaussian quantum channels. The codes are shown to be effective in protecting against errors that shift the values of the canonical variables q and p, and they can be used to implement fault-tolerant quantum computation and communication using harmonic oscillators or rotors. The paper also discusses the use of continuous-variable codes for many oscillators, and how they can be constructed from lattices in higher-dimensional phase space. The paper also discusses the use of concatenated codes to protect against a broader class of errors than small diffusive shifts applied to each oscillator. The paper concludes with a discussion of the physical realization of the coding schemes and the implications for quantum computing and communication.