ParametricPlot3D

ParametricPlot3D[{f_x,f_y,f_z},{u,u_min,u_max}]

produces a three-dimensional space curve parametrized by a variable u which runs from u_min to u_max.

ParametricPlot3D[{f_x,f_y,f_z},{u,u_min,u_max},{v,v_min,v_max}]

produces a three-dimensional surface parametrized by u and v.

ParametricPlot3D[{{f_x,f_y,f_z},{g_x,g_y,g_z}…}…]

plots several objects together.

ParametricPlot3D[…,{u,v}∈reg]

takes parameters {u,v} to be in the geometric region reg.

Details and Options

No curve or surface is drawn in any regions where the corresponding f_i or g_i evaluate to None, or anything other than real numbers.
ParametricPlot3D treats the variables u and v as local, effectively using Block.
ParametricPlot3D has attribute HoldAll, and evaluates the , , … only after assigning specific numerical values to variables.
In some cases it may be more efficient to use Evaluate to evaluate the , , … symbolically before specific numerical values are assigned to variables.
ParametricPlot3D has the same options as Graphics3D, with the following additions and changes:

Axes	True	whether to draw axes
BoundaryStyle	None	how to draw boundary lines for surfaces
ColorFunction	Automatic	how to determine the color of curves and surfaces
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every function evaluation
Exclusions	Automatic	u points or u,v curves to exclude
ExclusionsStyle	None	what to draw at excluded points or curves
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed
Mesh	Automatic	how many mesh divisions in each direction to draw
MeshFunctions	Automatic	how to determine the placement of mesh divisions
MeshShading	None	how to shade regions between mesh divisions
MeshStyle	Automatic	the style for mesh divisions
Method	Automatic	the method to use for refining surfaces
NormalsFunction	Automatic	how to determine effective surface normals
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotLegends	None	legends for surfaces
PlotPoints	Automatic	the initial number of sample points in each parameter
PlotRange	Automatic	range of values to include
PlotStyle	Automatic	graphics directives for the style for each object
PlotTheme	$PlotTheme	overall theme for the plot
RegionFunction	(True&)	how to determine whether a point should be included
TextureCoordinateFunction	Automatic	how to determine texture coordinates
TextureCoordinateScaling	True	whether to scale arguments to TextureCoordinateFunction
WorkingPrecision	MachinePrecision	the precision used in internal computations

Interactive labeling can be specified for curves or surfaces using Tooltip, StatusArea, or Annotation.
ParametricPlot3D[Tooltip[list],…] specifies that ${f_(x),f_(y),f_(z)}$ should be displayed as tooltip labels for the corresponding curves or surfaces.
Tooltip[{f_x,f_y,f_z},label] specifies an explicit tooltip label for a curve or surface.
All the functions etc. should give real numbers for all values of parameters at which they are evaluated. There will be holes in the final surface anywhere at which etc. do not yield real number values.
The default setting PlotPoints->Automatic corresponds to PlotPoints->75 for curves and PlotPoints->{15,15} for surfaces.
ParametricPlot3D initially evaluates each function at a number of equally spaced sample points specified by PlotPoints. Then it uses an adaptive algorithm to choose additional sample points, subdividing in each parameter at most MaxRecursion times.
You should realize that with the finite number of sample points used, it is possible for ParametricPlot3D to miss features in your functions. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion.
On[ParametricPlot3D::accbend] makes ParametricPlot3D print a message if it is unable to reach a certain smoothness of curve.
The default setting Mesh->Automatic corresponds to None for curves, and 15 for surfaces.
The default setting MeshFunctions->Automatic corresponds to {#4&} for curves, and {#4&,#5&} for surfaces.
The arguments supplied to functions in MeshFunctions and RegionFunction are x, y, z, u, and v. Functions in ColorFunction and TextureCoordinateFunction are by default supplied with scaled versions of these arguments.
The functions are evaluated all along each curve, or all over each surface.
By default, surfaces are treated as uniform white diffuse reflectors, corresponding to ColorFunction->(White&).
ParametricPlot3D returns Graphics3D[GraphicsComplex[data]].
Themes that affect 3D surfaces include: