We explore the constraints on those extensions to the standard models of cosmology and particle physics which modify the early-Universe, radiation dominated, expansion rate (parameterized by the effective number of neutrinos N_ν). The constraints on S(N_ν) and the baryon density parameter , derived from big bang nucleosynthesis (BBN, t~20 min), are compared with those inferred from the cosmic microwave background anisotropy spectrum (CMB, t~400 kyr) and large scale structure (LSS, t~14 Gyr). At present, BBN provides the strongest constraint on N_ν (N_ν = 2.4 ± 0.4 at 68% confidence), but a weaker constraint on the baryon density. In contrast, while the CMB/LSS data best constrain the baryon density (η₁₀ = 6.1_−0.1^+0.2 at 68% confidence), independent of N_ν, at present they provide a relatively weak constraint on N_ν which is, however, consistent with the standard value of N_ν = 3. When the best fit values and the allowed ranges of these CMB/LSS-derived parameters are used to calculate the BBN-predicted primordial abundances, there is excellent agreement with the observationally inferred abundance of deuterium and good agreement with ⁴He, confirming the consistency between the BBN and CMB/LSS results. However, the BBN-predicted abundance of ⁷Li is high, by a factor of 3 or more, if its observed value is uncorrected for possible dilution, depletion, or gravitational settling. We comment on the relation between the value of N_ν and a possible anomaly in the matter power spectrum inferred from observations of the Ly-α forest. Comparing our BBN and CMB/LSS results permits us to constrain any post-BBN entropy production as well as the production of any non-thermalized relativistic particles. The good agreement between our BBN and CMB/LSS results for N_ν and η_B permits us to combine our constraints finding, at 95% confidence, 1.8<N_ν<3.2 and 5.9<η₁₀<6.4.