We constrain six possible extensions to the Λ cold dark matter (CDM) model using measurements from the Dark Energy Survey's first three years of observations, alone and in combination with external cosmological probes. The DES data are the two-point correlation functions of weak gravitational lensing, galaxy clustering, and their cross-correlation. We use simulated data vectors and blind analyses of real data to validate the robustness of our results to astrophysical and modeling systematic errors. In many cases, constraining power is limited by the absence of theoretical predictions beyond the linear regime that are reliable at our required precision. The Λ CDM extensions are dark energy with a time-dependent equation of state, nonzero spatial curvature, additional relativistic degrees of freedom, sterile neutrinos with eV-scale mass, modifications of gravitational physics, and a binned σ8(z ) model which serves as a phenomenological probe of structure growth. For the time-varying dark energy equation of state evaluated at the pivot redshift we find (wp,wa)=(-0.9 9-0.17+0.28,-0.9 ±1.2 ) at 68% confidence with zp=0.24 from the DES measurements alone, and (wp,wa)=(-1.0 3-0.03+0.04,-0. 4-0.3+0.4) with zp=0.21 for the combination of all data considered. Curvature constraints of Ωk=0.0009 ±0.0017 and effective relativistic species Neff=3.1 0-0.16+0.15 are dominated by external data, though adding DES information to external low-redshift probes tightens the Ωk constraints that can be made without cosmic microwave background observables by 20%. For massive sterile neutrinos, DES combined with external data improves the upper bound on the mass meff by a factor of 3 compared to previous analyses, giving 95% limits of (Δ Neff,meff)≤(0.28 ,0.20 eV ) when using priors matching a comparable Planck analysis. For modified gravity, we constrain changes to the lensing and Poisson equations controlled by functions Σ (k ,z )=Σ0ΩΛ(z )/ΩΛ ,0 and μ (k ,z )=μ0ΩΛ(z )/ΩΛ ,0, respectively, to Σ0=0. 6-0.5+0.4 from DES alone and (Σ0,μ0)=(0.04 ±0.05 ,0.0 8-0.19+0.21) for the combination of all data, both at 68% confidence. Overall, we find no significant evidence for physics beyond Λ CDM .