3 continuous variables: make a row of such plots at 3-5 defined levels (or 'slices') of the third variable.

4 continuous variables: make a column of such plots at 3-5 defined levels of the fourth variable.

I would also consider using colour for the third dimension, though it makes it much harder to show the points in addition to the fitted surface. This can be partially ameliorated by showing an additional plot of observed vs. fitted values.

Here's an example from a [recently published paper] that shows the output of a theoretical model, but the same approach can be used for a statistical model. (https://aslopubs.onlinelibrary.wiley.com/doi/abs/10.1002/lno.12040):

In principle this plot matrix approach can be used with the 3D plots too, but it will likely become too crowded.

Categorical variable: same as above, it's just 2 panels (for a binary variable, more otherwise) instead of 3-5 levels as in the case of the continuous variable. Or if you use the 3D plots instead of colour, you may be able to squeeze 2-3 categories into the same panel using different colours.

What about 5 continuous variables? Animate each panel i.e. use time as the 5th dimension. It will rarely work well, but there will be a few datasets for which it's possible.

3 continuous variables: make a row of such plots at 3-5 defined levels (or 'slices') of the third variable.

4 continuous variables: make a column of such plots at 3-5 defined levels of the fourth variable.

I would also consider using colour for the third dimension, though it makes it much harder to show the points in addition to the fitted surface. This can be partially ameliorated by showing an additional plot of observed vs. fitted values.

Here's an example from a recently published paper that shows the output of a theoretical model, but the same approach can be used for a statistical model. (https://aslopubs.onlinelibrary.wiley.com/doi/abs/10.1002/lno.12040):

In principle this plot matrix approach can be used with the 3D plots too, but it will likely become too crowded.

Categorical variable: same as above, it's just 2 panels (for a binary variable, more otherwise) instead of 3-5 levels as in the case of the continuous variable. Or if you use the 3D plots instead of colour, you may be able to squeeze 2-3 categories into the same panel using different colours.

What about 5 continuous variables? Either make several sets of these plot matrices at 3-5 defined levels of the fifth variable, or animate each panel in the single plot matrix i.e. use time as the 5th dimension. These options will almost never work well, but there will be a few datasets for which it's possible.

3 continuous variables: make a row of such plots at 3-5 defined levels (or 'slices') of the third variable.

4 continuous variables: make a column of such plots at 3-5 defined levels of the fourth variable.

I would also consider using colour for the third dimension, though it makes it much harder to show the points in addition to the fitted surface. This can be partially ameliorated by showing an additional plot of observed vs. fitted values.

Here's an example from a recently published paper:

Categorical variable: same as above, it's just 2 panels instead of 3-5 levels as in the case of the continuous variable.

3 continuous variables: make a row of such plots at 3-5 defined levels (or 'slices') of the third variable.

4 continuous variables: make a column of such plots at 3-5 defined levels of the fourth variable.

I would also consider using colour for the third dimension, though it makes it much harder to show the points in addition to the fitted surface. This can be partially ameliorated by showing an additional plot of observed vs. fitted values.

Here's an example from a [recently published paper] that shows the output of a theoretical model, but the same approach can be used for a statistical model. (https://aslopubs.onlinelibrary.wiley.com/doi/abs/10.1002/lno.12040):

In principle this plot matrix approach can be used with the 3D plots too, but it will likely become too crowded.

Categorical variable: same as above, it's just 2 panels (for a binary variable, more otherwise) instead of 3-5 levels as in the case of the continuous variable. Or if you use the 3D plots instead of colour, you may be able to squeeze 2-3 categories into the same panel using different colours.

What about 5 continuous variables? Animate each panel i.e. use time as the 5th dimension. It will rarely work well, but there will be a few datasets for which it's possible.