Polygenic risk scores (PRS) based on European training data suffer reduced accuracy in non-European target populations, exacerbating health disparities. This loss of accuracy predominantly stems from LD differences, MAF differences (including population-specific SNPs), and/or causal effect size differences. PRS based on training data from the non-European target population do not suffer from these limitations, but are currently limited by much smaller training sample sizes. Here, we propose PolyPred, a method that improves cross-population polygenic prediction by combining two complementary predictors: a new predictor that leverages functionally informed fine-mapping to estimate causal effects (instead of tagging effects), addressing LD differences; and BOLT-LMM, a published predictor. In the special case where a large training sample is available in the non-European target population (or a closely related population), we propose PolyPred+, which further incorporates the non-European training data, addressing MAF differences and causal effect size differences. PolyPred and PolyPred+ require individual-level training data (for their BOLT-LMM component), but we also propose analogous methods that replace the BOLT-LMM component with summary statistic-based components if only summary statistics are available. We applied PolyPred to 49 diseases and complex traits in 4 UK Biobank populations using UK Biobank British training data (average N=325K), and observed statistically significant average relative improvements in prediction accuracy vs. BOLT-LMM ranging from +7% in South Asians to +32% in Africans (and vs. LDpruning + P-value thresholding (P+T) ranging from +77% to +164%), consistent with simulations. We applied PolyPred+ to 23 diseases and complex traits in UK Biobank East Asians using both UK Biobank British (average N=325K) and Biobank Japan (average N=124K) training data, and observed statistically significant average relative improvements in prediction accuracy of +24% vs. BOLT-LMM and +12% vs. PolyPred. The summary statistic-based analogues of PolyPred and PolyPred+ attained similar improvements. In conclusion, PolyPred and PolyPred+ improve cross-population polygenic prediction accuracy, ameliorating health disparities.