The emergence of SARS-CoV-2 variants threatens current vaccines and therapeutic antibodies and urgently demands powerful new therapeutics that can resist viral escape. We therefore generated a large nanobody repertoire to saturate the distinct and highly conserved available epitope space of SARS-CoV-2 spike, including the S1 receptor binding domain, N-terminal domain, and the S2 subunit, to identify new nanobody binding sites that may reflect novel mechanisms of viral neutralization. Structural mapping and functional assays show that indeed these highly stable monovalent nanobodies potently inhibit SARS-CoV-2 infection, display numerous neutralization mechanisms, are effective against emerging variants of concern, and are resistant to mutational escape. Rational combinations of these nanobodies that bind to distinct sites within and between spike subunits exhibit extraordinary synergy and suggest multiple tailored therapeutic and prophylactic strategies.