  We constrain extensions to the $\Lambda$CDM model using measurements from the
Dark Energy Survey's first three years of observations and external data. The
DES data are the two-point correlation functions of weak gravitational lensing,
galaxy clustering, and their cross-correlation. We use simulated data and blind
analyses of real data to validate the robustness of our results. In many cases,
constraining power is limited by the absence of nonlinear predictions that are
reliable at our required precision. The models are: dark energy with a
time-dependent equation of state, non-zero spatial curvature, sterile
neutrinos, modifications of gravitational physics, and a binned $\sigma_8(z)$
model which serves as a probe of structure growth. For the time-varying dark
energy equation of state evaluated at the pivot redshift we find $(w_{\rm p},
w_a)= (-0.99^{+0.28}_{-0.17},-0.9\pm 1.2)$ at 68% confidence with $z_{\rm
p}=0.24$ from the DES measurements alone, and $(w_{\rm p}, w_a)=
(-1.03^{+0.04}_{-0.03},-0.4^{+0.4}_{-0.3})$ with $z_{\rm p}=0.21$ for the
combination of all data considered. Curvature constraints of
$\Omega_k=0.0009\pm 0.0017$ and effective relativistic species $N_{\rm
eff}=3.10^{+0.15}_{-0.16}$ are dominated by external data. For massive sterile
neutrinos, we improve the upper bound on the mass $m_{\rm eff}$ by a factor of
three compared to previous analyses, giving 95% limits of $(\Delta N_{\rm
eff},m_{\rm eff})\leq (0.28, 0.20\, {\rm eV})$. We also constrain changes to
the lensing and Poisson equations controlled by functions $\Sigma(k,z) =
\Sigma_0 \Omega_{\Lambda}(z)/\Omega_{\Lambda,0}$ and $\mu(k,z)=\mu_0
\Omega_{\Lambda}(z)/\Omega_{\Lambda,0}$ respectively to
$\Sigma_0=0.6^{+0.4}_{-0.5}$ from DES alone and $(\Sigma_0,\mu_0)=(0.04\pm
0.05,0.08^{+0.21}_{-0.19})$ for the combination of all data. Overall, we find
no significant evidence for physics beyond $\Lambda$CDM.
