We present SPECULATOR - a fast, accurate, and flexible framework for emulating stellar population synthesis (SPS) models for predicting galaxy spectra and photometry. For emulating spectra, we use principal component analysis to construct a set of basis functions, and neural networks to learn the basis coefficients as a function of the SPS model parameters. For photometry, we parameterize the magnitudes (for the filters of interest) as a function of SPS parameters by a neural network. The resulting emulators are able to predict spectra and photometry under both simple and complicated SPS model parameterizations to percent-level accuracy, giving a factor of $10^3$-$10^4$ speed up over direct SPS computation. They have readily-computable derivatives, making them amenable to gradient-based inference and optimization methods. The emulators are also straightforward to call from a GPU, giving an additional order-of-magnitude speed-up. Rapid SPS computations delivered by emulation offers a massive reduction in the computational resources required to infer the physical properties of galaxies from observed spectra or photometry and simulate galaxy populations under SPS models, whilst maintaining the accuracy required for a range of applications.
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