Fit indices are the part of structural equation modelling that gets quoted most and understood least. A supervisor asks for them, a table appears in the results chapter, and the numbers sit there without anyone saying what they mean. Here is what each one is actually testing, and what none of them can do for you.
The chi-square is a test of exact fit
The model chi-square asks whether the covariance matrix your model implies differs from the one in your data. A non-significant result means no detectable difference.
In practice it is almost always significant once the sample passes a few hundred cases, because the test gains power to detect trivial differences. That is why nobody rejects a model on chi-square alone. Report it, with its degrees of freedom, and then read the approximate fit indices. Do not quietly leave it out because it was significant.
The indices you will be asked for
CFI and TLI compare your model against a null model in which nothing is related to anything. They run roughly from 0 to 1, and the commonly cited thresholds are .95 or above for good fit and .90 or above for acceptable.
RMSEA estimates error of approximation per degree of freedom, so it rewards parsimony. The usual cutoffs are .06 or below for good fit and up to .08 for reasonable. Report the 90 per cent confidence interval with it. A point estimate of .06 with an interval running to .11 is a much weaker claim than the single number suggests.
SRMR is the standardised difference between observed and predicted correlations. Below .08 is the common threshold.
Most of these cutoffs trace back to simulation work by Hu and Bentler in 1999. They were proposed as rules of thumb under particular conditions of sample size, estimator and model type, and they have been argued about ever since. Treat them as conventions your reviewers expect, not as laws.
What good fit does not mean
This is the part that matters, and it is where most vivas go wrong.
Good fit does not mean your theory is correct. It means the model you specified reproduces the covariances in your data acceptably well. For almost any model there are equivalent models, with different arrows and different stories, that fit exactly as well. Fit cannot choose between them. Your theory and your design have to.
Good fit does not rescue a weak measurement model. Check the measurement model first: factor loadings, composite reliability, average variance extracted, and discriminant validity between constructs. A structural model built on constructs that were not cleanly measured is a tidy result resting on nothing.
Good fit does not license the path you wanted. A model can fit well while the hypothesis at the centre of your study is not supported. Those are separate findings and both get reported.
Modification indices
When fit is poor, software will offer a list of changes that would improve it. Each one is a suggestion derived from your specific sample, and following them chases noise.
Correlating two error terms is defensible when the items share wording or method, and you say so and why. Adding a cross-loading because the number dropped is not defensible, and a careful examiner will ask you to justify it. If you make data-driven changes, report that you made them and treat the result as exploratory.
What to put in the chapter
Chi-square with degrees of freedom and p, CFI, TLI, RMSEA with its confidence interval, and SRMR. The estimator you used and why. The measurement model results before the structural ones. Any modification you made and the reason for it.
That is a table and two paragraphs, and it answers the questions an examiner is going to ask before they are asked.