The Wilkinson Microwave Anisotropy Probe (WMAP) 5-year data provide stringent limits on deviations from the minimal, six-parameter Λ cold dark matter model. We report these limits and use them to constrain the physics of cosmic inflation via Gaussianity, adiabaticity, the power spectrum of primordial fluctuations, gravitational waves, and spatial curvature. We also constrain models of dark energy via its equation of state, parity-violating interaction, and neutrino properties, such as mass and the number of species. We detect no convincing deviations from the minimal model. The six parameters and the corresponding 68% uncertainties, derived from the WMAP data combined with the distance measurements from the Type Ia supernovae (SN) and the Baryon Acoustic Oscillations (BAO) in the distribution of galaxies, are: Ω _b h ² = 0.02267^+0.00058 _–0.00059, Ω _c h ² = 0.1131 ± 0.0034, Ω_Λ = 0.726 ± 0.015, n_s = 0.960 ± 0.013, τ = 0.084 ± 0.016, and at k = 0.002 Mpc^-1. From these, we derive σ₈ = 0.812 ± 0.026, H ₀ = 70.5 ± 1.3 km s^-1 Mpc^–1, Ω _b = 0.0456 ± 0.0015, Ω _c = 0.228 ± 0.013, Ω _m h ² = 0.1358^+0.0037 _–0.0036, z _reion = 10.9 ± 1.4, and t ₀ = 13.72 ± 0.12 Gyr. With the WMAP data combined with BAO and SN, we find the limit on the tensor-to-scalar ratio of r < 0.22(95%CL), and that n_s > 1 is disfavored even when gravitational waves are included, which constrains the models of inflation that can produce significant gravitational waves, such as chaotic or power-law inflation models, or a blue spectrum, such as hybrid inflation models. We obtain tight, simultaneous limits on the (constant) equation of state of dark energy and the spatial curvature of the universe: –0.14 < 1 + w < 0.12(95%CL) and –0.0179 < Ω _k < 0.0081(95%CL). We provide a set of "WMAP distance priors," to test a variety of dark energy models with spatial curvature. We test a time-dependent w with a present value constrained as –0.33 < 1 + w ₀ < 0.21 (95% CL). Temperature and dark matter fluctuations are found to obey the adiabatic relation to within 8.9% and 2.1% for the axion-type and curvaton-type dark matter, respectively. The power spectra of TB and EB correlations constrain a parity-violating interaction, which rotates the polarization angle and converts E to B. The polarization angle could not be rotated more than –59 < Δα < 24 (95% CL) between the decoupling and the present epoch. We find the limit on the total mass of massive neutrinos of ∑m _ν < 0.67 eV(95%CL), which is free from the uncertainty in the normalization of the large-scale structure data. The number of relativistic degrees of freedom (dof), expressed in units of the effective number of neutrino species, is constrained as N _eff = 4.4 ± 1.5 (68%), consistent with the standard value of 3.04. Finally, quantitative limits on physically-motivated primordial non-Gaussianity parameters are –9 < f ^local _NL < 111 (95% CL) and –151 < f ^equil _NL < 253 (95% CL) for the local and equilateral models, respectively.