An important assessment prior to genome assembly and related analyses is genome profiling, where the k-mer frequencies within raw sequencing reads are analyzed to estimate major genome characteristics such as size, heterozygosity, and repetitiveness. Here we introduce GenomeScope 2.0 (https://github.com/tbenavi1/genomescope2.0), which applies combinatorial theory to establish a detailed mathematical model of how k-mer frequencies are distributed in heterozygous and polyploid genomes. We describe and evaluate a practical implementation of the polyploid-aware mixture model that quickly and accurately infers genome properties across thousands of simulated and several real datasets spanning a broad range of complexity. We also present a method called Smudgeplot (https://github.com/KamilSJaron/smudgeplot) to visualize and estimate the ploidy and genome structure of a genome by analyzing heterozygous k-mer pairs. We successfully apply the approach to systems of known variable ploidy levels in the Meloidogyne genus and the extreme case of octoploid Fragaria × ananassa.