In this paper, we define several ideal versions of Cauchy sequences and completeness in quasi-metric spaces. Some examples are constructed to clarify their relationships. We also show that: (1) if a quasi-metric space (X, ρ) is I-sequentially complete, for each decreasing sequence {F n } of nonempty I-closed sets with diam{F n } → 0 as n → ∞, then n∈N F n is a single-point set; (2) let I be a P-ideal, then every precompact left I-sequentially complete quasi-metric space is compact.