When the model fit is adequate, the model residuals should behave like white noise.
Check residual autocorrelations
For overall significance, and for significance at individual lags
Box-Pierce test, or Ljung-Box test
Look at the acf and pacf of model residuals
Model residuals should be iid N(0, s2)
The AR and MA terms are defined by the roots of the model polynomials. The roots of the polynomials should pass the Unit Root test.
An AR model should be stationary.
An MA model is always stationary, but should be invertible.