Dijet events are studied in the proton--proton collision dataset recorded at $\sqrt{s}=$13 TeV with the ATLAS detector at the Large Hadron Collider in 2015 and 2016, corresponding to integrated luminosities of 3.5 fb$^{-1}$ and 33.5 fb$^{-1}$ respectively. Invariant mass and angular distributions are compared to background predictions and no significant deviation is observed. For resonance searches, a new method for fitting the background component of the invariant mass distribution is employed. The dataset is then used to set upper limits at a 95% confidence level on a range of new physics scenarios. Excited quarks with masses below 6.0 TeV are excluded, and limits are set on quantum black holes, heavy W' bosons, W* bosons, and a range of masses and couplings in a Z' dark matter mediator model. Model-independent limits on signals with a Gaussian shape are also set, using a new approach allowing factorization of physics and detector effects. From the angular distributions, a scale of new physics in contact interaction models is excluded for scenarios with either constructive or destructive interference. These results represent a substantial improvement over those obtained previously with lower integrated luminosity.