We review late-time cosmic acceleration after DESI Data Release 2 (DR2), emphasizing the interplay between Type Ia supernovae (SNe Ia), anisotropic baryon acoustic oscillation (BAO), cosmic microwave background (CMB) calibration, and perturbation-sensitive probes (redshift-space distortions and weak lensing). DESI DR2 delivers percent-level BAO distance ratios over $0\lesssim z\lesssim 2.5$, including a $\mathrm{Ly}\alpha$-forest anchor at $z_{\mathrm{eff}}=2.33$. In CMB-calibrated combinations, flat $\mathrm{\Lambda }\mathrm{CDM}$ exhibits a mild parameter mismatch, while allowing evolving dark energy (e.g., Chevallier-Polarski-Linder $w_0–w_a$) can improve the fit; the preference is dataset-dependent and is particularly sensitive to redshift-dependent SN calibration/selection residuals at the $\mathrm{few}\times {10}^{-2}\mathrm{mag}$ level. To sharpen likelihood-level interpretation, we provide two diagnostics: (i) a BAO-shape observable independent of the drag-epoch sound horizon $r_d$, namely the Alcock-Paczynski (AP) combination $F_{\mathrm{AP}}(z)\equiv D_{\mathrm{M}}(z)/D_{\mathrm{H}}(z)$, constructed directly from published $(D_{\mathrm{M}}/r_d,D_{\mathrm{H}}/r_d)$ with covariance propagation; (ii) a linear-response map from SN Hubble-diagram systematics $\delta \mu (z)$ to induced biases in $(w_0,w_a)$, yielding an explicit calibration requirement for DESI-era claims of evolving $w(z)$. We synthesize parametric and nonparametric reconstructions of $w(z)$ and ${\rho }_{\mathrm{DE}}(z)$ and map the resulting phenomenology to microphysical dark-energy and modified-gravity models subject to perturbation stability and gravitational-wave propagation constraints.

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