Ten museums, most free or under €5, each addressing a different civilisation that built and dismantled itself on the same patch of Andalusian soil: Roman Corduba at its imperial height, the Umayyad Caliphate at its intellectual peak, Sephardic Jewish life before the 1492 expulsion, and a flamenco tradition that the city helped define. The Mezquita-Catedral gives you the physical proof of this layering, but the museums give you the specifics — what a 10th-century Caliphal palace interior looked like, what the Jewish scholars of the Judería actually studied, how Roman settlers lived in a city of 100,000 people before most of Europe had anything comparable.
The ten museums here are not ranked by size or fame. The ranking reflects what a visitor with limited time will find most useful and most distinctive. Several of the lower-ranked entries are, by any measure, more unusual than the top five: the guadamecí museum on number nine is probably the only place in the world where you can see a lost Umayyad craft recovered and practised. The bullfighting museum at number ten is polarising but occupies a 16th-century mansion in the Judería and tells a chapter of the city's identity that the archaeological record alone cannot.
Six of the ten museums on this list offer free entry, or free entry for EU citizens, or a free day each week. Córdoba's museum offer is genuinely affordable in a way that most European city equivalents are not. The three densest clusters are the Plaza del Potro (Bellas Artes, Julio Romero, Centro Flamenco Fosforito), the Judería (Casa de Sefarad, Casa Andalusí, Bullfighting Museum), and the Roman Bridge south bank (Museo Vivo de Al-Andalus). A third day in the city, after the Mezquita and Medina Azahara, is well spent working through these clusters on foot.
For context on the wider city: the top 10 monuments in Córdoba covers the Mezquita, Alcázar, and Medina Azahara in detail. Museums and monuments overlap in Córdoba, and the two lists read best together.