Causal inference is a critical step in improving our understanding of biological processes, and Mendelian randomization (MR) has emerged as one of the foremost methods to efficiently interrogate diverse hypotheses using large-scale, observational data from biobanks. Although many extensions have been developed to address the three core assumptions of MR-based causal inference (relevance, exclusion restriction, and exchangeability), most approaches implicitly assume that any putative causal effect is linear. Here, we propose PolyMR, an MR-based method that provides a polynomial approximation of an (arbitrary) causal function between an exposure and an outcome. We show that this method provides accurate inference of the shape and magnitude of causal functions with greater accuracy than existing methods. We applied this method to data from the UK Biobank, testing for effects between anthropometric traits and continuous health-related phenotypes, and found most of these (84%) to have causal effects that deviate significantly from linear. These deviations ranged from slight attenuation at the extremes of the exposure distribution, to large changes in the magnitude of the effect across the range of the exposure (e.g., a 1 kg/m ² change in BMI having stronger effects on glucose levels if the initial BMI was higher), to non-monotonic causal relationships (e.g., the effects of BMI on cholesterol forming an inverted U shape). Finally, we show that the linearity assumption of the causal effect may lead to the misinterpretation of health risks at the individual level or heterogeneous effect estimates when using cohorts with differing average exposure levels.