diff --git a/src/MAT_types.jl b/src/MAT_types.jl
index eff6f87..9b54c01 100644
--- a/src/MAT_types.jl
+++ b/src/MAT_types.jl
@@ -526,9 +526,11 @@ function from_datetime(obj::MatlabOpaque)
     return map_or_not(ms_to_datetime, dat)
 end
 
-# is the complex part the submilliseconds?
+# The imaginary part is the low half of a double-double: smaller than half an ulp of the
+# real part, so real + imag == real in Float64 and nothing is lost at DateTime resolution.
 ms_to_datetime(ms::Complex) = ms_to_datetime(real(ms))
 function ms_to_datetime(ms::Real)
+    isfinite(ms) || return missing  # NaT is stored as NaN; DateTime has no ±Inf either
     s, ms_rem = fldmod(ms, 1_000)  # whole seconds and remainder milliseconds
     return DateTime(1970, 1, 1) + Second(s) + Millisecond(round(Int, ms_rem))
 end
@@ -561,9 +563,13 @@ function from_duration(obj::MatlabOpaque)
     if isnothing(dat) || isempty(dat)
         return Millisecond[]
     end
-    return map_or_not(Millisecond, dat)
+    return map_or_not(ms_to_millisecond, dat)
 end
 
+# milliseconds are stored as Float64: rounded to whole ones, as datetimes are, and a
+# missing (NaN) or infinite duration is missing
+ms_to_millisecond(ms::Real) = isfinite(ms) ? Millisecond(round(Int, ms)) : missing
+
 to_matlab_millis(d::Millisecond) = Float64(Dates.value(d))
 
 function MatlabOpaque(d::ScalarOrArray{Millisecond})
diff --git a/test/missing_gen.m b/test/missing_gen.m
new file mode 100644
index 0000000..c368575
--- /dev/null
+++ b/test/missing_gen.m
@@ -0,0 +1,16 @@
+% Generates test/v7/missing_times.mat and test/v7.3/missing_times.mat: datetime and
+% duration arrays holding missing (NaT, NaN), infinite, or fractional-millisecond values.
+% Run from the test directory: matlab -batch "missing_gen"
+
+dt_nat = [datetime(2022, 7, 20, 12, 0, 0), NaT];
+dt_nat_scalar = NaT;
+dt_inf = [datetime(2022, 7, 20, 12, 0, 0), datetime(Inf, 'ConvertFrom', 'posixtime'), ...
+          datetime(-Inf, 'ConvertFrom', 'posixtime')];
+dt_nat_zoned = [datetime(2022, 7, 20, 12, 0, 0, 'TimeZone', 'Europe/London'), NaT('TimeZone', 'Europe/London')];
+dur_nan = [seconds(1.5), seconds(NaN)];
+dur_nan_scalar = seconds(NaN);
+dur_frac = milliseconds([1.5; 2.25; 2.75]);
+
+vars = {'dt_nat', 'dt_nat_scalar', 'dt_inf', 'dt_nat_zoned', 'dur_nan', 'dur_nan_scalar', 'dur_frac'};
+save(fullfile('v7', 'missing_times.mat'), vars{:}, '-v7');
+save(fullfile('v7.3', 'missing_times.mat'), vars{:}, '-v7.3');
diff --git a/test/read.jl b/test/read.jl
index 950480a..9b70045 100644
--- a/test/read.jl
+++ b/test/read.jl
@@ -275,6 +275,30 @@ for format in ["v7", "v7.3"]
     end
     end
 
+    # missing_times.mat is written by missing_gen.m: datetime and duration arrays holding
+    # missing (NaT, NaN), infinite or fractional-millisecond values
+    @testset "missing and fractional times $format" begin
+        filepath = joinpath(dirname(@__FILE__), format, "missing_times.mat")
+        if isfile(filepath)
+            # dt_nat_zoned has a time zone, which is dropped with a warning
+            vars = @test_logs (:warn, r"timezone") match_mode=:any matread(filepath)
+            noon = DateTime(2022, 7, 20, 12)
+
+            @test isequal(vars["dt_nat"], [noon missing])                 # NaT is missing
+            @test eltype(vars["dt_nat"]) == Union{Missing,DateTime}
+            @test ismissing(vars["dt_nat_scalar"])
+            @test isequal(vars["dt_inf"], [noon missing missing])         # so are ±Inf
+            @test isequal(vars["dt_nat_zoned"], [noon - Hour(1) missing]) # the UTC instant
+
+            @test isequal(vars["dur_nan"], [Millisecond(1500) missing])
+            @test ismissing(vars["dur_nan_scalar"])
+            @test vec(vars["dur_frac"]) == Millisecond.([2, 2, 3])       # 1.5, 2.25, 2.75 ms
+        else
+            # generated in MATLAB by test/missing_gen.m; skipped until it is committed
+            @test_skip isfile(filepath)
+        end
+    end
+
     @testset "user defined classdef $format" begin
     let objtestfile = "user_defined_classdefs.mat"
         filepath = joinpath(dirname(@__FILE__), format, objtestfile)