diff --git a/hypothesis_vcf/strategies.py b/hypothesis_vcf/strategies.py
index b8e4671..a8b1fa4 100644
--- a/hypothesis_vcf/strategies.py
+++ b/hypothesis_vcf/strategies.py
@@ -31,6 +31,7 @@ class Field:
     vcf_key: str
     vcf_type: str
     vcf_number: str
+    description: str = "Generated field"
 
     def get_header(self):
         return (
@@ -38,10 +39,20 @@ def get_header(self):
             f"ID={self.vcf_key},"
             f"Type={self.vcf_type},"
             f"Number={self.vcf_number},"
-            f'Description="{self.category},Type={self.vcf_type},Number={self.vcf_number}">'
+            f'Description="{self.description}">'
         )
 
 
+# GT is a special case, since it has a special syntax, and must be listed as the first
+# format field (if present)
+GT = Field(
+    category="FORMAT",
+    vcf_key="GT",
+    vcf_type="String",
+    vcf_number="1",
+    description="Genotype",
+)
+
 # references to the VCF spec are for https://samtools.github.io/hts-specs/VCFv4.3.pdf
 
 # [Table 1: Reserved INFO keys]
@@ -133,7 +144,7 @@ def vcf_numbers(category, max_number):
 def vcf_fields(category, max_number):
     # info flag fields must have number 0
     # non-flag fields can't have number 0
-    return builds(
+    general_fields = builds(
         Field,
         category=just(category),
         vcf_key=vcf_field_keys(category),
@@ -143,6 +154,11 @@ def vcf_fields(category, max_number):
         lambda field: (field.vcf_type == "Flag" and field.vcf_number == "0")
         or (field.vcf_type != "Flag" and field.vcf_number != "0")
     )
+    if category == "INFO":
+        return general_fields
+    else:
+        # FORMAT: GT special case
+        return one_of(just(GT), general_fields)
 
 
 # [1.6.1 Fixed fields]
@@ -183,8 +199,31 @@ def qualities():
     )
 
 
+# [1.6.2 Genotype fields]
+
+
+def genotypes(alleles, ploidy):
+    def gt_str(allele_indexes, phased):
+        sep = "|" if phased else "/"
+        return sep.join(
+            [str(idx) if idx is not None else "." for idx in allele_indexes]
+        )
+
+    return builds(
+        gt_str,
+        lists(
+            one_of(integers(0, alleles - 1), none()), min_size=ploidy, max_size=ploidy
+        ),
+        booleans(),
+    )
+
+
 @composite
 def vcf_values(draw, field, *, max_number, alt_alleles, ploidy):
+    # GT special case
+    if field is GT:
+        return [draw(genotypes(alleles=alt_alleles + 1, ploidy=ploidy))]
+
     # [1.3 Data types]
     if field.vcf_type == "Integer":
         # some integer values at lower end of range are not allowed
@@ -231,6 +270,15 @@ def vcf_number_to_ints(vcf_number, *, max_number, alt_alleles, ploidy):
     raise ValueError(f"Number '{vcf_number}' is not supported.")
 
 
+def ensure_gt_first(format_fields):
+    # GT must be the first field if present [1.6.2 Genotype fields]
+    try:
+        i = format_fields.index(GT)
+        format_fields.insert(0, format_fields.pop(i))
+    except ValueError:
+        pass
+
+
 @composite
 def vcf(
     draw,
@@ -290,6 +338,7 @@ def vcf(
             unique_by=lambda f: f.vcf_key.lower(),
         )
     )
+    ensure_gt_first(format_fields)
     sample_ids = draw(
         lists(
             text(alphabet=ALPHANUMERIC, min_size=1), max_size=max_samples, unique=True
diff --git a/tests/test_strategies.py b/tests/test_strategies.py
index b389e52..e2bcb3e 100644
--- a/tests/test_strategies.py
+++ b/tests/test_strategies.py
@@ -1,3 +1,5 @@
+import re
+
 import pysam
 from hypothesis import HealthCheck, given, note, settings
 from hypothesis.strategies import data
@@ -7,6 +9,7 @@
     RESERVED_FORMAT_KEYS,
     RESERVED_INFO_KEYS,
     Field,
+    genotypes,
     vcf_field_keys,
     vcf_fields,
     vcf_values,
@@ -40,6 +43,17 @@ def test_format_fields(data):
     assert field.vcf_number != "0"
 
 
+@given(data=data())
+def test_genotypes(data):
+    alleles = 3
+    ploidy = 2
+    gt = data.draw(genotypes(alleles=alleles, ploidy=ploidy))
+    allele_index_strs = re.split("[/|]", gt)
+    assert len(allele_index_strs) == ploidy
+    allele_indexes = [int(idx) if idx != "." else None for idx in allele_index_strs]
+    assert all(0 <= idx < alleles for idx in allele_indexes if idx is not None)
+
+
 @given(data=data())
 def test_vcf_values(data):
     field = Field("INFO", "I1", "Integer", "1")